## Ex-Increasing Set Sequence

Arrangements, sorting, packing, partitions, critical path analysis, networks, graphs, ...
Bubbler
Posts: 28
Joined: Sat Feb 23, 2013 6:12 am

### Ex-Increasing Set Sequence

Consider all subsets of {1,2,...,n}.
Pick some of the subsets and arrange them in increasing order by their average.
We call the sequence of sets Ex-Increasing Set Sequence of order n if:
1) there is no pair of sets in the sequence with same average, and
2) every pair of adjacent sets in the sequence is disjoint to each other.
Also, we define L(n) to be the length of longest possible Ex-Increasing Set Sequence of order n.

For example:
For n = 5, the sequence {1}, {2}, {1,4}, {3}, {2,5}, {4}, {5} satisfies both conditions, and it is the longest possible sequence.
So we have L(5) = 7.
Some other values are L(4) = 5, L(6) = 10, L(7) = 15.
For n = 7 the sequence is:
{1}, {2}, {1,4}, {3}, {1,2,7}, {3,4}, {1,2,5,7}, {4}, {1,3,6,7}, {4,5}, {1,6,7}, {5}, {4,7}, {6}, {7}
Can you find L(10), or even L(30)?

I coded a quite naive BFS in Python (a bit hard to read)
It does give a correct answer up to n=7(~300ms), but quickly gets out of scope from n=8.

Code: Select all

from time import clock
t0 = clock()
n = 7;
ar = reduce(lambda ar,x:[r.union(u) for u in [[],[x]] for r in ar],range(1,n+1),[set()])
ar.remove(set())
ar.sort(key=lambda x:(sum(x)+.0)/len(x))
maxlen = 0
maxseq = []
stack = [[ar[0]]]
while stack:
r = stack.pop()
j = ar.index(r[-1])
for i in xrange(j+1,len(ar)):
if r[-1].isdisjoint(ar[i]) and sum(r[-1]) * len(ar[i]) < sum(ar[i]) * len(r[-1]):
stack.append(r + [ar[i]])
if not updates and len(r) > maxlen:
maxseq, maxlen = r, len(r)
maxseq = [list(sorted(list(x))) for x in maxseq]
print maxlen, maxseq
print clock() - t0, 'sec'

quantheory
Posts: 7
Joined: Mon Jul 07, 2014 12:31 am

### Re: Ex-Increasing Set Sequence

My initial attempt at this had similar times. It seems that the number of sets that could go into a sequence is 2n - 1, and so if you then use a naive algorithm that's roughly exponential in the number of sets, the scales something like C^(2^n), which is pretty horrible.

My thought is that, if you can prove that the longest sequence (or one of the longest sequences, if they are not unique for some n) always contains each of the sets with a single number, then this would help quite a lot. Then you are guaranteed that {1}, {2}, ..., {n} are all in the sequence, and you only have to fill in intervals between these numbers, which decomposes the problem into pieces that should be far more manageable.

I haven't proven this yet, but I'm partway there. Let's say that you have a sequence for n, and it is missing the set {k} for some integer k. Then you have one of four possibilities:
1) There is no set with mean k in the sequence, and you can add the set {k} and still get a valid ex-increasing set sequence. Thus you have a new sequence that's one bigger.
2) There is no set with mean k, but you can't add the set {k}, because one of the sets on either side contains k. In that case, replace the problematic set with the set {k}; since the original set contained k, you know that neither of its neighbors has k. Therefore the new sequence is valid, and so you have a new sequence that's the same size.
3) There is a set with mean k, and it contains k (e.g. {3,4,5} has mean 4 and contains 4). Simply replace this set with {k}, and you have a new sequence that's the same size.
4) There is a set with mean k, and it does not contain k (e.g. {3,5} has mean 4 but does not contain 4). I believe that there's an algorithm that you can use to modify the sequence to contain {k} in all cases, without changing the number of elements, but I'm not quite there yet.

If (4) can be solved, then that would prove that for every sequence, you can convert it to a new sequence that contains all of the sets {1}, {2}, ..., {n}, and is equally long or longer, which would mean that you can restrict your search for longest sequences to these candidates.
thundre
Posts: 356
Joined: Sun Mar 27, 2011 10:01 am

### Re: Ex-Increasing Set Sequence

The obvious node-based algorithm uses O(2n) memory and O(4n) time at most. It runs very fast up to n=18.

Process each subset in order of average. Discard supersets with the same average.

Keep a list of subsets sorted by maximum position. For each subset you process, look back in the list to find the last non-intersecting subset. The current subset has maximum position one greater than that one.

L(1) = 1
L(2) = 2
L(3) = 3
L(4) = 5
L(5) = 7
L(6) = 10
L(7) = 15
L(8) = 21
L(9) = 29
L(10) = 39
L(11) = 49
L(12) = 63
L(13) = 79
L(14) = 99
L(15) = 121
L(16) = 145
L(17) = 171
L(18) = 203
L(19) = 237
L(20) = 277
L(21) = 321

Code: Select all

public class IncreasingSetSequence {

private static class Node {
private final float average;
private int value = 0;
/**
*/
int sum = 0;
long bit = 1;
int i = 1;
int count = 0;
if ((bit & mask) > 0) {
sum += i;
count++;
}
bit <<= 1;
i++;
}
average = sum / (float) count;
}
}

private final int n;

/**
* @param n
*/
public IncreasingSetSequence(int n) {
this.n = n;
}

public int calc() {
TreeMap<Float,HashSet<Node>> nodesByAverage = new TreeMap<Float,HashSet<Node>>();
long max = (1L << n);
HashSet<Node> set = nodesByAverage.get(n.average);
if (set == null) {
set = new HashSet<Node>();
nodesByAverage.put(n.average, set);
}
boolean ok = true;
// is there a subset of this node with the same average?
for (Node n0 : set) {
ok = false;
break;
}
}
if (ok) {
}
}

TreeMap<Integer,HashSet<Node>> nodesByValue = new TreeMap<Integer,HashSet<Node>>();
for (float average : nodesByAverage.keySet()) {
for (Node n1 : nodesByAverage.get(average)) {
int prevValue = 0;
if (!nodesByValue.isEmpty()) {
for (Integer v = nodesByValue.lastKey();
v != null && n1.value == 0; v = nodesByValue.lowerKey(v)) {
for (Node n0 : nodesByValue.get(v)) {
// n1 is a superset of n0, don't bother looking at lower v
n1.value = -1;
continue;
}
if (and != 0) continue;
if (n1.average == n0.average) continue;
n1.value = n0.value + 1;
break;
}
}
}
if (n1.value < 0) continue;
if (n1.value == 0)
n1.value = 1;
HashSet<Node> set = nodesByValue.get(n1.value);
if (set == null) {
set = new HashSet<Node>();
nodesByValue.put(n1.value, set);
}
}
}
int result = nodesByValue.lastKey();
System.out.println("L(" + n + ") = " + result);
return result;
}

public static void main(String args[]) {
for (int n = 1; n <= 30; n++) {
IncreasingSetSequence iss = new IncreasingSetSequence(n);
iss.calc();
}
System.exit(0);
}
}
thundre
Posts: 356
Joined: Sun Mar 27, 2011 10:01 am

### Re: Ex-Increasing Set Sequence

Here are some more results, with example sequences and the maximum size set in each example.

Code: Select all

L(1) = 1:  peak size 1 -- {1}
L(2) = 2:  peak size 1 -- {1} {2}
L(3) = 3:  peak size 1 -- {1} {2} {3}
L(4) = 5:  peak size 2 -- {1} {2} {1,4} {3} {4}
L(5) = 7:  peak size 2 -- {1} {2} {1,4} {3} {2,5} {4} {5}
L(6) = 10:  peak size 3 -- {1} {2} {1,4} {3} {1,4,5} {2,3,6} {4} {3,6} {5} {6}
L(7) = 15:  peak size 4 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,6,7} {4,5} {1,6,7} {5} {4,7} {6} {7}
L(8) = 21:  peak size 4 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {3,6} {1,5,8} {2,4,6,7} {5} {2,4,7,8} {5,6} {2,7,8} {6} {5,8} {7} {8}
L(9) = 29:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {2,3,5,7} {1,4,8} {3,6} {1,2,4,7,9} {3,5,6} {1,2,7,9} {5} {1,4,7,9} {3,5,8} {1,4,6,7,9} {3,8} {4,6,7} {1,5,8,9} {6} {2,8,9} {6,7} {3,8,9} {7} {6,9} {8} {9}
L(10) = 39:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {2,8} {1,3,6,7,9} {2,4,5,10} {1,6,9} {3,4,5,7,8} {2,9} {1,4,5,8,10} {2,6,9} {1,5,7,10} {2,4,6,8,9} {1,7,10} {3,5,8,9} {2,7,10} {4,5,6,8,9} {3,10} {4,7,9} {3,6,8,10} {7} {4,6,9,10} {7,8} {4,9,10} {8} {7,10} {9} {10}
L(11) = 49:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {5} {2,3,4,6,11} {1,5,7,8} {2,3,11} {1,4,5,7,10} {3,8} {2,4,5,7,10} {3,6,8} {2,5,7,9} {1,4,6,8,10} {2,7,9} {1,5,6,8,11} {2,4,9,10} {3,5,11} {2,4,7,9,10} {1,6,8,11} {3,4,7,9,10} {1,8,11} {5,6,7,9} {1,4,8,10,11} {5,9} {4,6,8,11} {5,7,10} {3,6,8,9,11} {5,10} {6,8,9} {3,7,10,11} {8} {4,10,11} {8,9} {5,10,11} {9} {8,11} {10} {11}
L(12) = 63:  peak size 7 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,9} {3,4,5,6,8} {1,2,7,11} {3,4,9} {1,2,6,7,11} {3,8} {1,5,6,7,9} {2,3,12} {1,4,8,10} {2,3,5,7,12} {1,4,9,10} {2,3,6,7,8,11} {1,4,5,9,12} {2,6,7,10} {1,3,4,5,8,11,12} {2,7,10} {1,5,6,8,12} {2,11} {1,3,8,9,12} {4,5,11} {1,2,6,7,9,10,12} {3,5,8,11} {2,4,7,9,12} {1,5,6,8,10,11} {7} {1,5,6,8,11,12} {3,7,9,10} {2,8,12} {3,4,9,10,11} {7,8} {1,5,9,11,12} {6,7,10} {2,8,9,12} {5,6,7,10,11} {3,9,12} {5,7,10,11} {4,9,12} {6,7,8,10,11} {5,12} {6,9,11} {5,8,10,12} {9} {5,11,12} {9,10} {6,11,12} {10} {9,12} {11} {12}
L(13) = 79:  peak size 7 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,9} {2,3,4,6,11} {1,5,7,8} {3,4,9} {1,2,6,7,11} {3,4,5,10} {1,2,6,7,12} {4,5,8} {2,3,6,12} {1,4,5,8,11} {2,10} {3,4,5,6,7,12} {1,2,8,9,11} {3,4,5,13} {1,2,6,7,8,9,11} {3,4,12} {1,5,7,9,10} {2,11} {1,5,6,7,8,9,10} {2,3,4,11,13} {5,7,8} {1,2,4,6,9,12,13} {3,5,8,11} {2,4,6,10,12} {1,3,7,8,9,13} {5,6,10} {2,3,4,9,12,13} {5,6,7,8,10} {2,3,11,13} {1,5,6,8,9,10,12} {2,7,13} {1,4,9,11,12} {3,5,6,7,8,10,13} {4,11} {1,7,8,10,12} {3,9,11} {1,5,6,7,10,12,13} {3,8,9,11} {2,5,7,12,13} {3,6,8,9,10,11} {4,7,13} {2,8,9,10,12} {4,5,11,13} {6,9,10} {2,4,11,12,13} {7,10} {5,6,8,11,13} {4,10,12} {5,8,9,13} {4,7,10,11,12} {9} {6,8,10,13} {7,9,12} {5,8,10,11,13} {7,12} {6,10,13} {7,9,11,12} {10} {7,9,12,13} {10,11} {7,12,13} {11} {10,13} {12} {13}
L(14) = 99:  peak size 8 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {3,7} {2,4,5,6,9} {1,3,7,10} {2,6,8} {1,3,4,9,10} {5,6} {1,2,4,8,13} {3,5,9} {1,2,6,14} {3,4,5,7,10} {1,11} {2,4,6,7,8,10} {1,3,5,9,13} {2,4,7,12} {1,3,5,6,8,10,11} {2,4,13} {1,3,7,9,12} {2,4,6,14} {1,3,5,7,9,10,11} {2,4,6,8,13} {1,5,14} {2,3,4,6,9,11,12} {1,5,8,13} {3,4,6,7,14} {1,2,8,9,10,11} {3,6,12} {1,2,7,8,9,10,13} {3,4,5,6,11,14} {1,2,8,12,13} {4,5,9,11} {1,3,6,7,8,12,14} {2,9,11} {1,3,4,7,8,10,12,14} {2,5,6,11,13} {1,3,4,8,10,12,14} {6,9} {1,3,5,7,11,12,14} {2,4,9,10,13} {1,3,5,7,8,11,12,14} {4,9,10} {1,3,5,6,12,13,14} {4,7,9,11} {1,5,6,13,14} {2,4,8,10,11,12} {1,5,6,7,9,13,14} {8} {1,6,7,10,11,14} {2,5,9,12,13} {3,6,10,14} {1,5,7,9,11,12,13} {3,8,14} {4,6,7,12,13} {2,5,8,9,10,11,14} {4,13} {6,7,9,10,11} {5,8,13} {2,6,7,9,11,12,14} {4,8,10,13} {3,6,9,12,14} {4,7,8,10,11,13} {9} {3,5,10,11,12,14} {7,8,9,13} {3,11,14} {7,8,9,10,13} {5,14} {6,8,10,11,13} {3,12,14} {7,9,10,13} {4,8,11,12,14} {10} {5,8,11,13,14} {9,10,12} {6,8,11,13,14} {9,12} {8,11,13} {7,10,12,14} {11} {7,13,14} {11,12} {8,13,14} {12} {11,14} {13} {14}
L(15) = 121:  peak size 9 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {5} {1,2,4,8,11} {3,5,6,7} {1,2,13} {3,4,5,7,8} {1,10} {3,4,6,7,8} {1,2,14} {3,4,7,9} {1,2,5,10,11} {3,9} {1,4,5,6,8,13} {2,3,7,9,10} {1,5,8,11} {2,3,4,6,7,9,13} {1,8,10} {2,3,6,7,14} {1,12} {2,4,5,6,8,10,11} {1,3,7,9,13} {2,6,12} {1,4,5,7,9,10,11} {2,3,8,14} {1,5,7,10,11} {2,3,6,8,9,13} {7} {1,3,4,5,10,12,15} {2,6,7,8,9,11} {1,3,5,13,14} {2,8,9,10} {1,3,5,6,11,12,13} {4,8,10} {1,2,3,6,9,11,12,15} {4,5,7,8,13} {1,3,6,9,10,11,12} {4,5,8,13} {2,3,6,9,10,11,12} {1,4,5,13,15} {2,3,6,7,8,10,11,14} {1,4,5,9,12,15} {2,3,7,8,10,11,13} {1,4,12,14} {2,3,5,6,7,9,10,13,15} {1,4,8,12,14} {3,5,6,7,11,15} {1,2,4,9,12,13,14} {3,10,11} {1,2,4,9,12,14,15} {5,6,7,8,10,13} {1,4,9,12,15} {2,3,6,7,8,10,11,13,14} {4,5,9,15} {1,3,6,10,11,13,14} {2,8,15} {3,5,6,7,9,11,12,14} {2,4,8,13,15} {3,5,7,9,10,11,14} {2,15} {1,6,7,8,11,13,14} {2,5,9,12,15} {3,4,6,8,10,11,13,14} {2,9,15} {1,5,7,10,11,13,14} {6,8,9,12} {3,7,10,11,13} {2,5,8,9,14,15} {3,6,7,10,11,12,13} {5,8,14} {1,3,11,12,13,15} {6,7,9,10,14} {1,8,13,15} {5,6,7,10,11,12,14} {4,9,15} {5,7,10,12,13} {3,6,8,9,11,14,15} {7,12} {3,9,10,11,15} {2,13,14} {3,8,9,10,11,12,15} {5,7,13,14} {4,9,10,11,15} {5,7,8,12,13,14} {9,11} {4,7,12,13,15} {6,10,11,14} {4,12,15} {7,8,10,13,14} {4,11,12,15} {7,9,10,13,14} {6,11,15} {8,9,12,14} {5,10,11,13,15} {8,14} {7,10,13,15} {9,11,14} {7,10,12,13,15} {9,14} {10,12,13} {7,11,14,15} {12} {8,14,15} {12,13} {9,14,15} {13} {12,15} {14} {15}
L(16) = 145:  peak size 9 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {2,7} {1,3,4,6,9} {2,5,7} {1,3,4,11} {2,8} {1,3,4,7,11} {2,5,6,8} {3,4,9} {1,2,5,7,12} {3,8} {1,2,5,9,11} {4,6,7} {1,3,5,14} {2,4,6,7,10} {1,3,5,15} {2,4,6,7,8,10} {1,3,5,9,13} {4,6,7,8} {1,2,3,5,9,10,14} {4,7,8} {2,3,6,10,11} {1,4,5,16} {2,3,6,7,8,9,11} {1,4,5,10,13} {3,8,9} {1,2,4,5,6,13,16} {3,7,8,9} {1,4,5,11,13} {2,3,6,7,8,15} {1,4,16} {2,3,5,8,9,10,13} {1,4,6,7,11,14} {2,5,8,9,12} {3,4,7,15} {1,2,5,6,11,12,14} {3,9,10} {1,2,5,6,8,11,12,14} {3,4,7,10,13} {1,2,6,8,9,12,14} {4,11} {1,2,5,7,9,13,16} {3,4,6,10,15} {1,2,5,8,9,11,12,13} {3,4,16} {2,5,6,7,9,12,13} {1,3,11,16} {2,4,5,6,7,8,10,13,15} {1,3,9,12,14} {2,5,6,8,10,16} {1,3,4,7,12,13,15} {2,14} {1,3,4,5,11,12,13,16} {2,6,7,8,9,10,15} {1,4,5,11,12,16} {2,7,8,10,14} {1,3,4,6,9,11,12,13,15} {2,7,8,16} {1,3,6,9,11,13,15} {4,5,16} {1,3,6,7,11,12,13,14} {2,5,9,10,16} {1,6,7,8,11,12,14} {2,9,10,13} {3,5,6,8,11,12,15} {1,4,9,13,16} {2,3,6,7,10,12,14,15} {5,8,13} {1,3,4,10,12,15,16} {6,7,8,14} {1,3,4,5,10,12,13,15,16} {2,8,9,11,14} {1,3,5,13,15,16} {4,6,8,9,10,11,14} {1,2,5,7,12,13,15,16} {6,10,11} {3,5,7,9,12,13,15} {1,6,8,10,14,16} {3,7,11,12,13} {2,4,5,8,9,10,14,15,16} {1,11,12,13} {3,4,8,9,10,15,16} {2,12,14} {1,5,8,9,10,11,15,16} {2,6,12,13,14} {3,4,8,9,11,15,16} {7,12} {4,5,6,9,13,14,16} {3,7,11,12,15} {2,5,8,9,10,13,14,16} {3,11,15} {4,6,9,10,12,13,14} {1,7,15,16} {2,10,11,12,14} {6,7,8,9,13,16} {2,5,10,11,12,14,15} {8,9,13} {3,6,7,14,15,16} {5,10,11,12,13} {4,7,14,16} {5,6,10,11,12,13,15} {1,14,16} {8,9,10,12,13} {4,6,7,11,14,15,16} {8,13} {5,6,12,14,16} {4,13,15} {6,7,9,11,12,14,16} {5,10,13,15} {4,8,12,14,16} {7,9,10,11,13,15} {8,14} {7,9,12,13,15} {8,10,11,16} {5,14,15} {7,9,12,13,16} {6,11,14,15} {7,10,12,13,16} {9,11,15} {8,12,13,14} {7,10,11,15,16} {12} {8,11,14,16} {10,12,15} {8,11,13,14,16} {10,15} {9,13,16} {10,12,14,15} {13} {9,15,16} {13,14} {10,15,16} {14} {13,16} {15} {16}
L(17) = 171:  peak size 9 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,3,11} {2,4,5,7,8} {1,3,6,11} {2,4,10} {1,3,5,7,11} {2,9} {1,3,4,5,7,8,11} {2,6,9} {1,4,8,10} {2,3,5,7,12} {1,4,9,10} {2,3,6,7,8,11} {1,4,5,9,12} {2,3,6,14} {1,4,5,7,8,9,10} {2,6,11} {1,3,5,10,13} {2,4,8,12} {1,3,5,7,9,10,11} {2,4,6,8,13} {1,9,10} {2,3,4,6,7,8,17} {1,5,9,12} {2,4,7,8,13} {1,3,5,6,9,17} {2,4,8,14} {1,3,5,6,9,11,15} {2,4,7,8,10,12} {1,3,5,11,16} {2,6,7,14} {1,3,4,9,10,11,13} {2,6,14} {3,4,5,7,8,9,10,13} {1,2,6,12,16} {3,4,5,7,8,10,15} {6,9} {2,3,4,7,8,13,16} {1,5,6,11,15} {2,3,4,7,8,9,12,16} {1,5,6,10,11,13} {2,3,4,8,9,12,16} {1,5,10,15} {2,3,4,6,7,9,11,12,16} {1,5,8,10,15} {2,4,6,9,12,14} {1,3,5,8,10,13,15} {2,4,9,17} {1,3,5,6,10,11,13,16} {2,4,7,8,9,12,15} {1,5,6,10,11,16} {3,4,8,9,17} {1,2,5,6,7,10,13,14,16} {3,4,11,15} {1,5,7,9,10,12,14} {4,8,13} {1,2,5,6,9,12,15,17} {3,4,8,13,14} {2,5,6,9,10,11,16} {3,4,12,15} {2,5,6,7,9,10,11,13,14} {1,3,4,8,12,15,17} {2,7,9,11,14} {1,4,5,6,8,12,16,17} {2,11,13} {1,3,4,10,12,14,17} {2,9,11,13} {1,3,4,7,8,10,14,15,17} {2,5,9,12,16} {1,4,6,11,14,17} {2,5,8,9,10,12,16} {1,3,4,6,7,13,14,15,17} {5,8,11,12} {3,4,6,7,10,13,14,16} {2,5,8,9,11,12,17} {3,4,6,13,14,15} {2,7,9,12,16} {3,4,5,6,10,11,13,14,17} {1,9,12,15} {3,5,6,10,11,14,16} {1,2,8,13,15,17} {3,4,5,10,11,12,14,16} {2,7,8,13,17} {4,5,6,9,11,15,16} {1,3,7,8,10,12,13,14,17} {5,6,11,16} {1,2,9,10,13,15,17} {5,7,8,12,16} {2,4,6,9,10,14,15,17} {1,12,16} {2,5,6,11,13,14,17} {8,9,10,12} {1,3,5,6,11,14,15,16,17} {7,8,9,12,13} {1,2,10,14,15,17} {6,7,8,9,11,12,16} {2,3,5,10,13,14,15,17} {1,11,12,16} {4,6,9,10,13,14,15} {3,5,8,12,16,17} {2,10,11,13,15} {4,5,7,8,9,12,14,16,17} {2,11,13,15} {6,7,8,9,12,14,16} {3,13,15} {2,7,8,10,11,12,16,17} {4,6,13,14,15} {1,8,9,10,12,16,17} {4,11,13,14} {1,7,8,10,15,16,17} {5,9,12,13,14} {3,6,7,10,11,15,16,17} {5,13,14} {4,8,9,11,12,15,16} {6,10,13,14} {2,8,11,16,17} {6,7,10,13,14,15} {3,8,9,11,12,16,17} {5,13,15} {7,8,9,12,14,17} {6,10,11,13,16} {5,8,15,17} {6,9,10,11,13,14,16} {2,15,17} {9,10,11,13,14} {5,7,8,12,15,16,17} {9,14} {6,8,10,12,13,15,17} {5,14,16} {4,10,11,12,13,15,17} {8,9,14,16} {5,10,12,15,17} {8,9,11,13,14,16} {4,15,17} {8,10,13,14,16} {6,11,15,17} {9,12,16} {7,10,13,15,17} {8,12,14,16} {7,11,13,15,17} {8,14,16} {9,12,13,17} {8,11,14,15,16} {10,12,17} {8,13,14,15,16} {11,12,17} {9,13,14,15,16} {10,17} {11,14,16} {10,13,15,17} {14} {10,16,17} {14,15} {11,16,17} {15} {14,17} {16} {17}
L(18) = 203:  peak size 11 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {2,8} {1,3,6,7,9} {2,4,5,10} {1,7,8} {2,4,5,6,10} {1,3,7,11} {2,4,5,8,9} {1,3,13} {2,4,7,10} {1,5,6,8,9} {2,4,7,11} {1,3,5,6,9,13} {2,4,7,8,10} {1,5,6,13} {2,3,4,7,8,9,11} {1,6,12} {2,3,5,8,14} {6,7} {1,3,4,5,9,11,13} {2,6,7,8,10} {3,4,13} {1,2,5,8,9,10,12} {3,4,6,14} {2,5,7,8,12} {1,3,4,6,10,11,13} {2,5,14} {1,4,6,7,9,10,13} {2,3,5,8,11,14} {4,6,7,9,10} {2,3,8,16} {1,4,5,7,9,11,14} {2,3,17} {1,4,5,6,7,8,13,15} {2,3,9,11,12} {1,4,5,6,7,14,15} {2,3,8,17} {1,4,6,7,9,12,14} {2,3,5,13,15} {1,4,6,8,9,10,11,12} {3,7,13} {1,2,6,9,10,12,14} {3,5,7,16} {1,2,4,8,9,10,11,12,13} {3,5,6,7,18} {1,4,8,9,12,13} {2,3,5,6,7,11,14,15} {4,12} {2,3,6,8,9,10,11,16} {1,4,5,7,12,13,15} {2,3,8,9,11,16} {1,5,6,12,17} {2,3,4,7,8,9,11,14,16} {1,5,10,17} {2,3,7,8,11,13,14} {4,5,16} {2,3,6,8,9,12,13,14} {4,5,7,11,15} {2,3,8,9,10,13,14} {4,5,7,18} {1,2,3,8,10,11,12,13,17} {4,5,6,7,9,14,15} {1,2,10,12,18} {4,5,6,7,8,9,13,17} {1,2,3,12,16,18} {5,6,7,8,10,11,14} {2,4,12,17} {1,3,6,7,8,9,13,14,18} {2,5,10,12,15} {3,6,8,9,11,16} {1,2,4,10,12,15,18} {3,5,6,8,9,11,13,16} {2,4,12,18} {1,3,6,7,9,10,14,15,17} {2,4,5,8,11,12,13,18} {1,3,6,7,14,16,17} {2,4,9,10,12,18} {1,3,5,6,7,8,11,13,14,16,17} {2,4,10,12,18} {1,5,6,7,8,11,14,15,16} {2,10,12,13} {3,4,5,9,11,15,18} {1,2,6,7,8,10,12,14,16,17} {3,4,5,11,15,18} {1,6,7,9,10,12,13,17} {3,4,11,14,15} {2,6,8,9,12,13,16} {1,4,5,7,10,11,14,15,18} {6,13} {1,2,4,9,10,11,15,16,18} {3,6,7,8,12,14,17} {2,5,10,13,18} {1,3,4,9,12,15,16,17} {2,13,14} {1,3,4,7,9,11,12,15,17,18} {2,5,8,10,13,14,16} {3,6,12,18} {1,4,5,7,11,13,14,16,17} {2,8,9,12,18} {1,3,5,6,7,11,13,14,15,16,17} {2,8,9,10,12,18} {3,5,7,11,13,14,16} {1,4,6,8,10,15,17,18} {2,5,7,9,11,12,13,14,16} {1,4,17,18} {3,6,7,9,11,14,15,16} {1,5,8,10,12,17,18} {3,4,11,13,14,16} {7,8,9,12,15} {2,3,4,10,11,13,14,17,18} {1,9,15,16} {3,6,10,11,12,13,17} {7,9,15} {1,2,8,11,13,14,16,18} {6,7,10,12,17} {2,5,8,13,14,15,16} {4,6,7,9,10,11,12,17,18} {3,8,15,16} {4,5,6,11,13,17,18} {7,8,10,12,16} {2,4,9,11,13,14,15,17} {1,7,10,12,16,18} {4,6,9,11,13,15,17} {3,10,14,16} {2,5,7,9,11,13,15,17,18} {4,8,12,14,16} {2,6,11,13,15,18} {3,7,9,10,14,16,17} {4,6,8,11,12,13,15,18} {7,10,16} {1,4,9,14,15,17,18} {7,8,11,12,13,16} {6,9,10,14,17} {4,5,7,11,12,13,15,16,18} {6,8,14,17} {4,7,10,11,13,16,18} {2,15,17} {5,7,9,11,12,13,16,18} {3,8,14,15,17} {5,7,9,12,13,16,18} {8,15} {3,6,9,12,16,17,18} {5,11,13,14,15} {3,8,9,10,12,16,17,18} {7,13,15} {4,8,11,12,14,16,17} {1,13,15,18} {4,8,14,16,17} {5,9,11,13,15,18} {6,8,10,12,14,16,17} {4,11,15,18} {8,9,10,12,13,16,17} {3,11,14,15,18} {4,12,16,17} {7,9,10,13,14,15,18} {4,16,17} {6,10,13,15,18} {8,9,11,12,14,16,17} {4,13,15,18} {8,10,11,12,14,16,17} {7,13,18} {8,10,11,12,15,16,17} {6,13,14,18} {10,11,12,15,16} {6,9,13,14,17,18} {11,15} {7,9,12,16,17,18} {11,13,14,15} {5,17,18} {9,13,14,15,16} {7,12,17,18} {10,13,14,15,16} {6,17,18} {12,13,14,16} {9,10,15,17,18} {14} {10,12,17,18} {13,14,16} {10,12,15,17,18} {13,16} {12,15,17} {11,14,16,18} {15} {11,17,18} {15,16} {12,17,18} {16} {15,18} {17} {18}
L(19) = 237:  peak size 11 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,4,6,8} {5} {1,3,4,7,11} {2,5,6,8} {3,4,9} {1,2,5,6,13} {3,8} {1,5,6,7,9} {3,4,10} {1,6,7,9} {2,3,4,8,12} {6} {1,3,4,7,9,13} {2,5,6,8,10} {1,3,7,14} {2,4,5,6,8,9,10} {1,7,11} {2,3,4,5,6,8,10,13} {1,12} {2,3,4,5,9,10,13} {1,6,7,8,11} {2,5,13} {1,3,4,6,7,10,16} {2,5,9,11} {1,3,4,8,18} {2,5,6,7,9,12} {3,4,14} {1,2,5,8,9,10,15} {3,4,6,7,11,12} {1,2,8,9,16} {5,6,7,11} {1,3,4,8,9,10,16} {2,6,14} {1,3,4,5,7,9,13,17} {2,6,8,10,11} {1,3,4,5,9,13,17} {7,8} {1,3,4,6,9,12,18} {2,5,8,10,13} {1,3,4,6,7,9,15,16} {5,8,10} {2,3,6,7,9,12,15} {1,5,8,17} {2,3,4,6,9,10,11,12,13} {1,5,7,8,18} {3,4,6,9,10,15} {1,2,5,7,11,12,17} {8} {1,4,5,6,9,11,13,16} {2,3,7,8,10,12,15} {1,4,6,9,11,18} {2,3,10,12,14} {1,4,6,7,8,9,11,13,15} {2,5,12,14} {1,3,6,9,10,13,16} {2,5,18} {1,3,4,6,7,13,14,19} {2,5,8,9,18} {1,4,6,10,11,12,15} {3,5,8,18} {1,2,4,7,10,11,13,14,15} {3,5,6,8,9,12,17} {1,4,10,13,15} {2,3,5,8,9,12,14,16} {4,7,15} {1,2,3,6,8,10,11,13,16,17} {4,5,7,19} {1,2,3,6,11,12,13,15,16} {4,5,8,9,18} {1,2,10,11,13,16} {3,4,6,8,12,14,15} {1,2,5,9,10,11,16,17} {3,6,8,19} {1,4,5,9,10,11,13,14,15} {2,3,6,7,8,12,17,18} {1,4,5,10,13,15,16} {2,6,7,11,12,17} {1,3,4,5,8,9,10,13,14,16,18} {2,6,7,12,19} {3,4,5,8,9,11,13,14,16} {1,2,15,19} {3,5,8,9,11,13,16} {1,2,4,6,7,10,12,15,17,19} {3,5,8,11,13,16} {1,2,6,7,10,12,18,19} {5,8,9,11,14} {1,4,6,10,13,15,17} {2,3,5,7,8,12,14,16,18} {1,9,13,15} {3,4,6,7,10,11,12,16,17} {1,2,5,9,13,18,19} {3,7,10,12,16} {1,2,4,6,13,14,18,19} {5,9,15} {1,2,3,6,8,11,13,16,18,19} {4,5,7,9,12,14,17} {1,2,3,6,8,10,11,13,16,18,19} {4,9,12,14} {2,5,6,7,8,11,13,17,19} {3,4,9,15,18} {1,2,5,6,8,11,12,13,14,17,19} {3,4,9,10,15,18} {1,2,5,12,13,17,19} {4,7,8,9,10,11,14,16} {1,2,3,6,12,13,15,18,19} {5,7,11,17} {1,3,6,8,9,14,15,16,19} {2,4,7,10,11,12,17,18} {3,6,8,9,14,15,16} {2,5,11,12,13,18} {1,4,6,7,8,9,10,15,16,17,19} {3,5,11,14,18} {1,2,4,8,12,13,16,17,19} {3,5,15,18} {1,6,7,8,9,10,11,12,14,16,19} {2,3,4,13,15,17,18} {1,6,7,8,9,11,12,14,16,19} {3,4,5,15,17,18} {6,7,8,9,10,11,13,19} {1,2,14,17,18} {4,6,7,10,11,16,19} {2,3,5,8,12,14,15,17,18} {1,6,16,19} {2,3,8,11,12,13,14,15,17} {1,4,6,10,16,18,19} {5,8,12,13,15} {3,4,9,10,11,14,16,18} {5,8,19} {3,4,7,9,10,12,13,14,17,18} {1,5,8,11,15,16,19} {7,10,12,14} {1,2,8,9,11,13,16,18,19} {5,6,12,14,17} {2,4,7,8,9,10,11,15,16,18,19} {3,6,12,13,14,17} {1,4,8,11,15,18,19} {3,5,9,10,13,14,16,17} {2,6,7,8,11,12,15,18,19} {4,13,16} {2,7,9,10,12,14,17,18} {3,6,8,11,15,16,19} {1,5,12,14,17,18} {6,9,10,15,16} {1,7,8,11,12,13,14,17,18} {4,10,15,16} {2,8,11,12,13,14,19} {5,6,9,15,16,17} {4,7,8,10,12,13,18,19} {1,9,14,16,17} {6,7,10,11,13,15,18} {3,5,8,9,12,14,16,17,19} {7,10,11,18} {3,5,6,15,16,17,19} {2,11,13,14,18} {5,6,9,10,12,15,17,19} {2,7,13,14,16,18} {4,6,10,11,15,17,19} {7,9,13,18} {1,4,8,12,14,15,16,17,19} {7,10,11,13,18} {1,5,14,15,17,19} {7,8,9,12,13,16,18} {3,6,10,11,14,15,17,19} {8,9,13,18} {5,7,11,14,15,16,17} {2,9,12,13,18,19} {3,10,15,16,17} {6,8,9,11,12,13,14,18,19} {7,10,15,17} {5,9,11,13,14,16,18} {3,15,19} {5,8,10,11,14,16,17,18} {3,12,13,15,19} {5,8,9,14,16,17,18} {3,13,15,19} {5,6,12,14,16,17,18} {7,9,13,15,19} {5,8,11,12,14,16,17,18} {10,13,15} {3,9,11,14,16,17,19} {8,10,15,18} {6,12,13,16,17} {9,10,11,14,15,18} {6,7,12,13,16,17,19} {5,14,15,18} {9,10,11,13,17,19} {5,12,15,16,18} {9,11,14,19} {2,12,13,15,16,17,18} {10,11,19} {7,9,16,17,18} {10,11,12,13,14,15,19} {3,16,17,18} {8,12,14,15,19} {7,16,18} {8,10,13,14,15,17,19} {9,12,16,18} {7,11,15,17,19} {10,12,13,14,16,18} {9,19} {7,12,15,16,17,18} {11,13,14,19} {10,15,18} {11,12,14,16,19} {8,15,17,18} {11,13,14,16,19} {9,17,18} {10,14,16,19} {11,13,15,17,18} {10,16,19} {11,15,17,18} {13,14,19} {11,15,16,17,18} {12,19} {14,16,17} {11,15,18,19} {16} {13,15,18,19} {16,17} {13,18,19} {17} {16,19} {18} {19}
L(20) = 277:  peak size 12 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,6,8} {2,3,4,7,10} {1,5,6,9} {2,3,11} {1,4,5,7,10} {2,9} {1,3,4,8,12} {2,5,10} {1,3,6,13} {2,4,5,7,8,9} {1,3,14} {2,4,5,6,9,11} {1,3,7,8,12} {2,4,5,14} {1,3,6,7,8,9,10} {2,4,13} {1,6,7,8,10} {2,3,4,17} {1,5,6,7,8,9,10} {2,3,4,11,13} {1,7,12} {2,3,4,6,8,9,15} {1,5,7,14} {2,4,8,9,11} {1,3,6,7,10,14} {2,4,5,8,16} {1,3,6,7,10,11,12} {2,4,5,8,9,15} {1,6,7,10,12} {2,3,5,19} {1,4,6,7,8,9,16} {3,5,14} {1,2,4,6,8,9,11,18} {3,5,7,10,12} {1,2,4,6,9,14,16} {7,8} {1,2,4,6,9,11,20} {3,5,7,10,13} {1,2,4,6,8,9,15,16} {5,7,11} {1,2,6,8,10,13,14} {3,4,9,15} {1,2,5,6,7,8,10,13,18} {3,4,9,11,12} {2,6,7,8,10,14} {1,3,4,5,11,15,16} {2,14} {1,3,4,8,9,12,13,15} {2,5,6,7,10,11,16} {1,3,4,9,13,19} {2,5,6,12,16} {1,3,4,7,8,9,11,13,18} {2,6,10,15} {1,3,8,9,11,12,14} {5,7,13} {1,2,3,4,9,12,16,20} {5,6,7,10,14} {1,2,4,8,9,15,20} {3,5,10,16} {1,4,6,7,8,9,11,14,17} {2,3,5,10,12,13,15} {1,4,7,11,20} {2,3,5,6,8,13,14,18} {1,10,15} {3,4,6,7,9,14,18} {1,2,12,20} {3,4,5,7,8,10,11,15,16} {1,2,9,13,19} {4,5,8,10,12,14} {1,3,7,9,11,13,18} {2,4,5,6,8,14,15,17} {1,7,19} {2,3,5,8,10,11,12,14,17} {1,4,6,7,9,13,15,18} {2,3,5,11,12,14,17} {1,4,9,10,15,16} {2,3,5,6,7,8,11,12,13,14,20} {1,4,10,15,16} {2,3,6,7,8,9,11,18,19} {1,5,15,16} {3,4,6,7,10,17,18} {1,2,5,8,9,11,12,14,15,16} {3,4,6,10,13,20} {1,2,5,7,8,9,11,12,14,15,19} {3,4,6,16,18} {1,5,7,9,12,13,19} {2,3,4,6,10,11,14,17,18} {1,8,9,20} {2,3,4,6,10,13,14,16,18} {1,7,8,9,11,12,19} {2,5,6,15,20} {1,4,7,8,12,13,14,18} {5,6,9,10,11,17} {1,2,4,7,8,12,14,15,16,18} {3,5,6,10,11,13,20} {1,2,4,7,8,9,12,14,15,17,18} {3,5,11,20} {2,4,8,9,10,12,13,14,16} {5,6,7,11,20} {2,3,4,8,9,10,12,13,14,15,18} {1,5,6,11,16,20} {2,7,8,9,12,13,18} {1,3,5,10,11,14,15,20} {2,4,6,7,9,12,13,17,19} {1,5,16,18} {2,3,4,7,9,11,14,15,17,19} {1,5,6,8,10,12,13,16,20} {2,4,7,9,11,14,15,19} {1,5,6,10,12,17,20} {3,7,8,11,13,19} {1,4,5,6,9,10,12,14,15,16,20} {2,8,11,13,17} {1,3,4,9,10,12,16,18,19} {6,7,13,15} {1,2,3,5,8,10,14,16,17,18,19} {4,6,7,9,11,15,20} {1,3,5,8,10,12,13,16,17,18} {2,7,9,11,14,19} {1,3,4,5,8,10,12,16,17,18,20} {2,6,7,9,11,14,15,19} {1,4,12,17,18} {3,5,6,7,8,9,10,11,13,14,19,20} {1,2,4,15,16,17,18} {3,6,8,9,10,11,13,14,20} {4,17} {3,5,6,7,11,12,15,16,20} {1,2,4,13,17,18,19} {3,5,6,7,8,10,11,12,14,15,16,20} {1,4,13,17,18} {3,5,7,10,12,14,15,19} {1,2,4,6,9,11,13,16,17,18,20} {3,10,19} {1,4,5,8,9,11,15,16,18,20} {3,6,7,10,13,17,19} {1,2,4,5,11,12,14,15,16,18,20} {3,10,13,17} {1,2,6,9,11,14,15,19,20} {3,5,12,16,18} {1,2,6,7,9,10,13,15,17,19,20} {4,8,11,12,14,16} {1,6,7,9,15,18,20} {2,3,10,11,12,13,17,19} {1,6,7,8,9,14,15,18,20} {2,4,5,10,11,12,13,16,17,19} {6,7,20} {2,4,9,10,13,14,15,16,17} {1,3,5,11,12,18,19,20} {4,8,9,10,14,16,17} {1,6,7,15,18,20} {3,5,8,9,10,11,12,13,16,17,19} {1,6,14,15,20} {2,5,8,9,11,12,17,18,19} {4,10,15,16} {1,5,6,7,11,12,13,14,17,18,20} {2,8,9,10,15,16,19} {1,5,6,7,12,13,14,17,18,20} {3,9,10,11,16,19} {1,5,7,12,14,15,17,20} {8,9,11,13,16} {2,5,10,12,14,17,20} {4,6,8,9,11,13,15,18,19} {1,5,12,14,17,20} {2,4,8,9,13,15,16,18,19} {1,6,11,12,14,17,20} {5,9,13,15,16} {1,6,7,8,14,18,19,20} {3,10,11,13,16,17} {4,5,6,7,9,14,15,18,19,20} {3,10,11,12,13,16,17} {5,9,15,18} {2,3,4,11,14,16,17,19,20} {1,12,13,15,18} {2,6,7,8,10,11,14,16,17,19,20} {4,9,12,13,15,18} {3,8,10,11,14,17,20} {2,7,9,12,13,15,18,19} {5,6,8,10,11,14,16,17,20} {4,13,19} {7,8,9,11,12,14,16,20} {3,5,10,13,17,18,19} {6,7,9,15,16,20} {1,12,13,17,18} {4,7,9,10,11,14,16,19,20} {6,8,17,18} {4,7,9,13,14,19,20} {1,10,12,16,17,18} {6,7,9,11,13,14,19,20} {5,10,12,17,18} {7,8,11,13,14,15,19} {4,6,9,10,12,16,17,18,20} {7,11,13,19} {3,6,10,15,16,18,20} {8,11,12,13,19} {4,7,10,14,15,16,17,18} {5,9,11,12,19,20} {4,8,13,14,15,17,18} {5,10,16,20} {4,6,9,13,14,15,17,18,19} {5,11,12,16,20} {1,10,14,15,18,19} {5,9,11,12,16,17,20} {6,8,10,13,14,15,18,19} {7,12,20} {6,9,10,15,16,17,19} {4,11,12,14,18,20} {6,8,16,17,19} {7,9,11,12,13,14,15,18,20} {1,16,17,19} {8,10,11,12,14,18,20} {5,16,19} {6,9,12,13,14,15,18,20} {5,10,16,17,19} {8,9,11,13,15,18,20} {2,16,17,19} {5,10,13,14,15,18,20} {9,11,12,17,19} {5,8,13,14,15,16,18,20} {10,12,19} {1,13,14,15,16,17,20} {6,12,18,19} {3,13,16,17,20} {9,10,12,15,18,19} {6,11,13,14,16,17,20} {5,18,19} {10,11,12,13,16,17,20} {5,14,15,18,19} {10,11,16,20} {6,12,13,15,17,18,19} {7,16,20} {11,13,14,15,19} {8,10,12,16,17,18,20} {11,13,15,19} {5,12,14,16,17,18,20} {10,15,19} {11,12,13,14,16,17,20} {7,15,18,19} {11,13,14,16,20} {8,12,15,17,18,19} {11,13,16,20} {10,12,15,17,18,19} {11,14,16,20} {9,18,19} {12,13,15,17,20} {9,16,18,19} {12,14,15,17,20} {10,18,19} {11,15,17,20} {12,14,16,18,19} {13,15,20} {11,16,17,18,19} {14,15,20} {12,16,17,18,19} {13,20} {15,17,18} {12,16,19,20} {17} {13,19,20} {17,18} {14,19,20} {18} {17,20} {19} {20}
L(21) = 321:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,5,9} {1,4,10} {2,3,6,7,8} {1,4,5,11} {3,6,7} {1,2,5,8,11} {3,4,6,9} {1,2,5,8,12} {4,6,7} {2,3,8,10} {1,4,5,7,12} {2,10} {1,4,5,6,9,12} {2,3,7,8,11} {1,5,6,13} {2,3,4,7,8,9,11} {1,6,12} {2,4,5,7,14} {1,3,9,13} {2,4,5,6,8,10,11} {1,3,7,9,13} {4,6,10} {1,2,3,5,8,9,19} {4,6,7,10} {1,2,3,8,20} {4,5,6,7,9,10} {1,13} {3,4,5,6,8,9,15} {1,2,7,10,11,12} {3,4,6,8,15} {1,2,7,19} {3,4,5,6,8,12,13} {1,7,14} {2,3,4,5,6,10,11,18} {1,7,8,9,12} {2,3,4,5,6,15,17} {1,14} {2,3,4,6,9,12,17} {1,5,8,10,14} {2,3,4,6,7,9,11,19} {1,10,12} {3,4,5,6,7,9,20} {1,2,11,17} {3,4,5,6,7,8,9,13,15} {1,2,10,12,14} {3,5,6,9,11,13} {1,2,4,7,10,12,19} {3,5,16} {1,4,6,7,9,10,13,15} {2,3,5,8,11,12,16} {1,4,6,7,14,17} {2,3,8,10,18} {1,4,5,6,7,9,11,15,16} {2,3,8,20} {1,4,5,7,10,15,16} {3,8,14} {1,2,4,6,11,12,15,16} {3,5,7,8,19} {1,2,4,9,10,13,20} {5,6,7,16} {1,2,4,9,10,11,12,13,15} {3,5,6,7,8,14,17} {1,2,9,13,18} {3,4,6,7,8,11,14,16} {1,2,9,10,12,18} {3,4,6,7,11,13,17} {2,5,8,20} {1,3,4,6,9,10,13,15,18} {2,5,11,12,14} {3,4,8,9,13,16} {1,2,6,10,11,15,17} {3,4,5,7,8,9,16,19} {2,6,10,18} {3,4,5,7,8,9,11,16,19} {1,2,6,10,12,13,14,15} {3,5,7,8,9,11,21} {1,2,4,12,16,20} {3,5,6,7,8,9,10,11,13,14,15} {1,2,4,18,21} {3,5,6,7,8,11,12,15,16} {1,2,13,21} {4,5,7,9,10,12,18} {1,2,3,6,8,11,13,14,16,19} {4,5,7,10,12,18} {2,3,6,8,9,14,16,17} {1,5,11,12,18} {2,4,6,8,13,14,19} {1,3,5,9,10,11,12,16,18} {2,17} {1,3,4,8,12,13,14,15,16} {2,5,6,7,10,17,20} {1,3,11,14,19} {2,4,5,8,9,13,15,21} {7,10,12} {1,2,4,5,8,9,11,18,19,20} {3,6,7,10,12,14,16} {1,2,4,5,8,9,11,13,17,18,19} {3,7,14,15} {1,4,5,6,9,12,13,17,21} {2,3,10,16,18} {1,4,6,7,8,11,12,13,14,15,17} {2,3,5,10,18,21} {4,6,9,11,12,13,14} {1,2,7,8,10,16,17,18} {3,4,5,6,9,13,14,15,20} {2,7,21} {1,3,6,8,9,11,13,15,16,19} {2,4,5,7,10,12,14,17,20} {1,3,8,9,11,13,15,21} {2,4,7,10,12,17,19} {1,3,9,11,16,21} {2,5,6,7,8,10,12,13,14,15,20} {3,4,9,16,19} {1,2,5,8,11,13,14,17,21} {4,9,10,18} {1,3,5,6,8,12,13,14,15,17,19} {2,4,7,9,11,18,21} {1,3,5,8,10,12,14,15,16,19} {2,11,18} {3,4,5,7,8,9,10,15,16,17,20} {1,2,6,11,12,14,18,19} {3,5,10,13,21} {1,2,4,6,7,9,11,15,16,17,18,19} {3,5,8,10,13,14,20} {1,2,4,6,12,15,16,17,21} {3,8,11,20} {2,5,6,7,9,10,13,14,15,17,18} {1,3,4,8,11,12,16,19,21} {2,6,7,9,15,17,18} {1,3,4,5,8,10,11,12,13,19,20,21} {2,6,14,15,16} {4,8,9,10,11,12,13,18} {1,2,3,5,6,7,16,17,19,20,21} {4,9,10,12,14,15} {1,2,3,5,8,13,16,18,20,21} {4,7,9,11,12,15,17} {1,2,5,6,8,10,13,14,18,20,21} {3,11,12,17} {1,5,7,8,10,13,15,18,20} {3,9,11,14,17} {1,2,4,6,8,12,13,15,18,19,21} {5,9,10,11,14,16} {1,2,3,6,7,8,12,13,15,17,18,19,20} {4,5,9,10,11,16,21} {1,2,3,12,13,17,19,20} {4,5,7,8,9,10,16,18,21} {1,2,3,6,12,14,15,17,19,20} {5,10,13,16} {1,2,3,9,11,12,15,18,19,21} {4,5,6,8,10,14,16,17,20} {1,2,7,11,13,15,19,21} {3,5,9,10,16,17,18} {1,2,4,6,7,11,12,13,14,15,19,20,21} {5,8,9,10,17,18} {2,4,6,7,11,12,13,14,15,19,20} {3,5,9,18,21} {1,2,4,10,13,15,17,19,20} {3,9,12,21} {2,4,5,8,10,13,14,15,16,17,20} {3,7,9,11,12,18,19} {1,2,4,8,10,14,16,17,20,21} {9,12,13} {1,4,5,6,7,11,15,16,19,20,21} {2,8,9,10,13,14,17,18} {4,5,12,16,20} {1,2,6,7,8,10,13,15,17,18,19,21} {3,4,11,12,14,16,20} {1,5,7,9,10,15,17,18,21} {3,4,6,8,11,12,13,14,16,19,20} {2,7,10,15,17,18} {3,5,8,9,11,13,14,20,21} {2,6,7,12,17,18,19} {3,4,5,8,9,11,13,14,15,16,20,21} {6,7,10,17,18} {2,3,11,12,14,15,16,20} {1,5,7,8,9,10,13,17,18,19,21} {4,15,16} {1,2,7,8,12,13,14,19,20,21} {5,6,10,11,15,17,18} {1,3,7,8,9,12,13,16,19,20,21} {6,10,14,17} {1,4,5,7,11,18,19,20,21} {6,9,12,15,17} {1,2,5,7,8,13,16,18,19,20,21} {4,10,11,14,15,17} {5,6,7,12,13,19,21} {1,4,10,14,15,16,17,18} {2,5,7,9,11,13,19,20,21} {3,6,8,10,12,14,15,16,17,18} {2,5,20,21} {1,8,9,10,12,14,15,16,17,19} {3,4,7,11,13,18,20,21} {8,9,10,12,14,15,17} {4,5,11,13,19,21} {2,3,8,9,12,14,15,16,17,18,20} {4,6,11,19,21} {2,7,8,10,12,16,17,18,20} {6,9,13,21} {5,7,8,10,11,12,14,15,16,17,20} {2,4,9,13,18,19,21} {5,6,8,10,12,14,15,16,17,20} {4,7,11,13,18,21} {2,5,6,10,12,14,15,16,17,19,20} {3,7,13,18,21} {8,10,11,12,14,15,17} {2,3,6,7,16,18,19,20,21} {8,13,14,15} {2,6,7,9,11,18,19,20,21} {1,12,13,14,15,16,17} {2,3,18,19,21} {6,9,10,11,14,15,16,20} {8,12,18} {4,6,9,10,11,14,16,17,19,21} {3,8,12,13,15,18,20} {6,10,16,19} {2,8,9,12,13,15,17,18,21} {5,6,14,19,20} {3,8,9,10,11,13,15,16,17,18,21} {5,7,12,14,19,20} {1,6,13,15,16,18,21} {3,9,10,11,14,17,19,20} {6,7,8,12,13,15,16,18,21} {5,10,11,19,20} {1,8,12,14,15,16,18,21} {6,7,10,13,17,19,20} {5,8,12,15,18,21} {7,9,13,17,20} {2,8,10,12,14,15,18,19,21} {9,11,13,20} {2,5,7,12,14,15,16,17,18,19,21} {9,11,20} {7,8,12,13,15,16,17,19} {2,10,14,20,21} {7,9,12,15,16,17,18} {5,8,10,11,13,14,19,20,21} {9,12,16,17} {2,6,8,13,15,18,19,20,21} {9,12,14,16,17} {1,6,11,13,18,19,20,21} {12,14,15} {4,9,11,16,17,18,21} {1,15,19,20} {8,9,10,11,14,16,17,18,21} {2,13,15,19,20} {7,11,14,16,17,18} {6,9,10,12,19,20,21} {7,11,13,14,15,16,17,18} {3,12,20,21} {7,10,13,15,17,18,19} {5,9,14,16,20,21} {10,12,13,17,19} {6,7,11,14,15,16,18,20,21} {8,13,17,19} {5,9,14,15,16,20,21} {7,17,19} {5,6,14,15,16,18,20,21} {11,12,13,17,19} {5,10,14,15,16,20,21} {12,17} {7,10,13,15,18,19,20} {9,12,14,17,21} {8,10,11,15,16,18,19,20} {13,14,17} {9,11,12,15,16,19,21} {4,17,18,20} {9,11,14,19,21} {10,13,15,16,17,18} {7,11,12,14,19,20,21} {9,16,17,18} {10,12,13,15,20,21} {8,14,17,18,19} {10,15,16,20} {5,13,14,17,18,19,21} {10,16,20} {9,11,17,19,21} {12,13,14,15,16,18,20} {5,17,19,21} {10,14,16,18,20} {11,15,21} {12,13,14,16,17,18,20} {8,15,19,21} {11,14,16,18,20} {10,13,15,17,19,21} {14,18} {9,15,17,19,21} {11,16,18,20} {13,15,21} {9,16,18,19,20} {12,21} {13,14,17,19,20} {11,18,21} {15,16,17,19} {11,14,18,20,21} {17} {14,16,19,20} {13,18,21} {15,16,17,19,20} {14,21} {15,18,20} {14,17,19,21} {18} {15,17,20,21} {18,19} {15,20,21} {19} {18,21} {20} {21}
L(22) = 369:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {5} {2,3,4,6,11} {1,5,7,8} {2,4,10} {1,3,6,8,9} {4,7} {1,2,6,8,11} {3,4,10} {1,2,6,14} {3,4,5,8,9} {2,10} {1,3,5,7,8,13} {2,4,6,9,10} {1,3,7,14} {2,4,5,6,8,9,10} {1,3,15} {2,4,5,7,14} {1,12} {2,3,4,5,9,10,13} {1,6,7,8,11} {3,4,13} {2,5,6,7,8,9,10} {1,3,4,19} {2,5,6,8,13} {1,3,7,9,10,11} {2,4,6,16} {1,3,5,8,9,11,13} {2,4,6,7,10,14} {3,5,8,9,11} {1,6,10,12} {2,3,5,7,9,11,14} {1,8,13} {2,3,4,5,7,9,14,15} {1,6,8,10,12} {2,3,4,5,7,9,22} {1,14} {3,4,6,7,8,9,16} {1,2,5,12,18} {3,4,6,7,8,9,11,13} {1,10,12} {2,3,5,8,9,13,14} {1,4,7,19} {2,3,5,6,8,9,10,11,16} {1,4,7,13,14} {2,3,6,8,11,17} {1,4,5,7,9,10,12,15} {2,3,6,21} {1,4,5,7,9,11,12,16} {2,3,6,8,10,13,15} {1,4,5,7,12,20} {3,6,8,9,15} {1,2,4,5,7,10,11,12,22} {3,8,9,13} {2,4,5,6,10,15,16} {1,3,7,12,13,14} {2,4,5,6,8,9,16,17} {1,3,10,13,15} {2,4,5,7,11,14,16} {1,3,13,17} {2,4,5,7,9,10,11,14,15} {1,3,6,8,12,13,17} {2,5,7,9,20} {1,3,4,6,10,12,16,17} {2,11,13} {1,3,4,5,6,8,12,15,16,17} {2,9,11,13} {1,3,4,5,6,7,15,17,21} {2,9,10,11,12} {3,4,7,8,13,18} {1,2,5,9,14,15,16} {3,4,6,7,8,10,11,13,18} {1,2,5,12,15,19} {3,4,6,8,9,10,11,13,18} {1,2,5,7,12,14,15,17} {3,4,6,9,10,11,21} {2,5,7,8,16,17} {1,3,4,6,9,10,11,12,13,14,18} {2,5,7,15,17} {1,3,4,9,10,11,12,13,20} {2,6,7,22} {1,4,5,10,14,15,16} {2,3,6,7,8,9,11,12,13,22} {4,10,14} {2,3,6,7,12,13,15,17} {1,5,9,10,22} {3,4,7,11,12,14,15} {1,2,5,6,9,10,13,17,22} {3,7,8,20} {1,4,5,6,9,12,13,15,21} {2,3,7,10,11,16,18} {1,8,9,13,17} {3,4,5,6,12,14,15,18} {7,9,13} {1,2,3,5,6,10,16,17,18,19} {4,7,8,9,12,13,15} {1,2,3,5,6,10,11,14,16,17,22} {4,7,9,19} {2,3,5,6,11,12,13,16,20} {1,7,8,14,19} {2,3,4,5,6,10,11,13,16,17,21} {1,8,9,12,14,15} {4,5,6,7,10,16,21} {1,3,8,11,12,13,14,17} {2,4,5,7,9,10,15,18,19} {1,6,13,20} {2,3,5,8,10,11,12,15,17,18} {1,4,6,7,9,13,14,16,21} {2,3,5,8,11,12,18,22} {1,4,6,9,14,16,21} {3,5,8,11,12,22} {1,2,4,6,7,9,10,16,18,19,20} {3,5,8,13,22} {1,2,4,6,9,12,17,20,21} {3,5,11,22} {1,2,4,6,7,8,14,15,17,19,20} {3,5,9,10,11,13,21} {1,2,4,6,8,12,14,17,19,20} {3,9,10,11,13,16} {2,4,5,6,7,8,12,15,17,18,20} {1,3,9,10,11,14,16,19} {2,4,7,18,21} {1,3,5,6,8,9,10,13,14,17,19,20} {2,4,7,11,12,16,21} {1,5,6,8,10,14,15,17,18} {3,4,7,13,16,20} {1,2,5,6,9,10,12,15,17,18,21} {3,4,7,8,11,13,14,16,19} {1,2,5,10,17,18,21} {3,4,6,7,8,9,12,13,14,15,16,20} {1,2,11,18,21} {3,4,7,10,12,14,16,19} {1,2,5,6,8,11,13,15,17,18,21} {4,9,19} {1,3,5,7,10,11,13,16,20,21} {2,4,8,9,12,18,22} {1,3,6,7,10,11,13,14,16,17,20} {2,4,18,19} {1,5,6,9,10,12,13,20,21} {2,4,14,15,19} {1,3,5,8,9,10,11,13,16,21,22} {2,6,7,14,17,19} {1,3,4,5,8,9,10,11,12,15,20,21,22} {2,6,7,13,14,16,18} {1,5,8,9,10,11,21,22} {2,4,6,7,12,14,15,18,20} {1,5,8,9,10,11,13,16,17,19} {2,20} {3,5,6,7,9,10,13,14,15,19,21} {1,2,4,8,11,12,16,17,18,22} {3,5,6,7,10,14,15,19,21} {1,4,8,9,12,16,17,22} {2,5,7,11,13,19,21} {1,3,4,6,8,9,10,14,15,16,17,20,22} {2,5,11,12,18,19} {1,3,4,6,7,13,14,16,17,20,22} {2,9,12,15,18} {1,4,5,10,11,14,16,19,21} {3,6,8,9,12,15,17,20} {1,2,5,7,10,11,13,14,18,21,22} {4,6,9,12,15,16,17} {1,2,3,10,11,13,14,18,20,21} {4,5,9,12,16,22} {1,2,3,6,10,14,15,17,18,19,20} {4,5,9,11,12,13,16,21} {1,2,3,6,7,8,10,15,17,18,19,20,22} {4,9,11,12,21} {1,2,6,7,8,10,13,16,17,18,19,20} {3,4,5,11,14,21,22} {1,2,6,12,13,15,17,18,19} {3,4,5,7,9,10,11,14,20,21,22} {8,15} {1,3,4,6,7,11,16,18,19,20,22} {2,5,8,10,12,14,15,17,21} {4,6,9,11,13,18,20} {1,2,3,5,7,12,14,16,17,19,21,22} {8,9,10,13,18} {1,2,3,4,7,11,12,14,15,19,20,21,22} {5,6,8,10,13,16,17,18} {1,2,3,9,11,12,14,15,19,20,22} {4,5,7,16,17,21} {2,3,9,10,11,12,14,15,19,22} {4,5,6,8,18,20,21} {1,3,9,10,11,12,13,15,16,17,22} {2,6,19,20} {3,5,8,11,12,13,16,17,21} {6,7,10,14,22} {1,2,3,8,12,13,15,17,18,20,21} {5,9,10,11,14,22} {1,2,3,6,7,12,13,15,17,18,19,20,21} {4,8,9,10,14,16,22} {3,5,6,12,13,17,19,20} {1,4,9,10,11,14,15,21,22} {2,5,7,8,12,13,16,17,19,20} {1,4,6,9,10,11,14,15,18,21,22} {2,5,16,17,20} {3,7,8,9,11,12,13,15,21,22} {2,5,6,10,14,16,17,19,20} {3,7,8,9,12,15,21,22} {1,5,13,14,16,17,19} {2,4,6,8,9,10,11,12,15,18,20,21,22} {3,5,13,16,17,19} {1,4,7,8,10,11,12,18,20,21,22} {3,9,13,17,19} {1,5,6,8,11,16,20,21,22} {3,12,15,19} {1,5,6,9,10,13,14,16,18,21,22} {3,8,11,12,15,17,20} {4,5,7,9,10,14,16,18,19,21} {3,8,11,13,17,22} {1,5,6,7,9,14,16,18,19,20,21} {4,8,10,11,12,15,17,22} {1,6,14,20,21} {4,5,9,10,11,12,13,15,16,17,18,19} {1,2,7,14,20,21,22} {5,6,8,11,12,16,17,18,19} {2,4,7,9,10,13,14,15,20,21,22} {1,11,12,16,17,18} {3,5,6,8,14,15,19,21,22} {1,10,11,12,16,18,20} {3,4,6,8,9,13,14,15,17,19,21,22} {2,11,12,18,20} {1,3,13,14,15,16,17,22} {5,6,8,9,10,11,12,18,19,20,21} {2,14,22} {1,8,9,10,13,15,16,17,18,20} {5,6,11,12,14,19,22} {3,7,8,9,10,13,15,16,18,20,21} {2,5,11,12,14,17,19,22} {3,6,7,9,15,16,18,20,21} {5,11,12,17,19} {2,6,7,8,10,13,14,18,20,21,22} {4,9,12,16,17,19} {2,6,7,14,18,21,22} {3,5,10,13,16,17,19,20} {6,7,8,9,11,14,18,21,22} {3,4,10,12,13,15,16,17,19,20} {1,11,14,18,21} {4,7,8,10,13,15,19,20,22} {3,9,11,12,14,17,18,21} {6,7,8,10,19,20,22} {4,12,13,14,15,21} {2,5,6,9,11,16,17,18,19,20,22} {8,10,13,14,21} {1,5,6,12,16,18,19,20,22} {8,9,15,21} {3,7,10,11,12,13,14,17,18,19,22} {4,8,9,15,16,20,21} {2,6,10,12,13,14,17,18,19,22} {5,7,11,16,20,21} {2,4,13,14,15,18,19,22} {8,9,12,17,21} {1,4,14,16,18,19,22} {2,10,11,12,13,15,17,20,21} {1,16,18,19} {2,5,8,13,14,17,20,21,22} {9,10,11,12,16,18,19} {5,6,14,21,22} {3,7,11,15,16,18,19,20} {4,8,13,14,21,22} {1,9,10,12,15,16,17,18,19,20} {7,8,11,13,14,21,22} {4,12,19,20} {1,10,11,14,15,16,17,18,22} {7,9,13,19,21} {6,8,10,11,12,14,15,16,18,20,22} {4,9,13,17,19,21} {5,11,12,14,15,18,22} {4,8,10,13,16,19,20,21} {7,9,11,12,14,15,17,18,22} {4,10,13,16,20,21} {6,8,11,12,17,18,19,22} {3,10,14,15,16,20,21} {2,11,13,18,19,22} {8,12,15,16,20} {5,6,11,13,14,17,19,21,22} {3,16,18,20} {5,8,11,12,13,14,15,17,19,21,22} {9,16,18} {2,10,12,13,17,19,20,22} {3,14,16,18,21} {6,8,9,17,19,20,22} {10,11,12,13,14,15,16,18,21} {1,8,17,19,20,22} {11,12,13,14,15,16,21} {5,7,19,20,22} {3,13,14,15,16,17,18,21} {6,10,11,19,20,22} {7,12,14,15,16,18,21} {5,13,19,22} {10,11,12,14,15,16,17,18,20} {3,9,19,21,22} {7,13,15,16,18,20} {10,11,12,14,17,19,21} {6,9,13,15,16,18,20,22} {4,14,17,19,21} {7,10,13,16,18,20,22} {8,12,14,17,19,21} {10,13,15,16,22} {5,11,12,14,17,18,19,20,21} {10,13,16,22} {6,8,15,18,19,20,21} {11,13,22} {7,12,14,15,16,18,20,21} {9,10,17,19,22} {8,11,14,16,18,20,21} {12,19} {4,11,14,17,20,21,22} {10,15,16,18,19} {9,11,12,13,17,20,21,22} {14,15,18} {8,12,13,16,19,20,22} {7,17,18,21} {6,15,16,20,22} {12,13,14,17,18,21} {9,10,15,16,19,20,22} {8,17,18,21} {13,14,15,16,19,20} {8,12,18,21,22} {10,16,19,20} {7,14,15,17,18,21,22} {13,16,20} {10,12,17,21,22} {13,14,15,16,18,19,20} {6,17,21,22} {14,15,16,18,20} {11,17,22} {12,14,15,16,19,20,21} {10,17,18,22} {12,15,16,20,21} {11,14,17,18,19,22} {15,16,20} {11,13,19,21,22} {14,17,18,20} {12,19,21} {14,15,16,20,22} {12,18,19,21} {14,15,17,20,22} {13,19,21} {14,17,18,22} {13,16,19,20,21} {18} {11,19,21,22} {17,18,20} {14,16,19,21,22} {17,20} {16,19,21} {15,18,20,22} {19} {16,18,21,22} {19,20} {16,21,22} {20} {19,22} {21} {22}
L(23) = 419:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,9} {3,4,5,6,8} {1,2,7,11} {3,4,9} {1,2,5,8,11} {4,7} {1,2,3,9,13} {4,5,8} {2,3,7,11} {1,4,5,9,10} {2,3,7,12} {1,4,5,6,8,13} {2,3,7,9,10} {1,5,8,11} {2,3,4,6,7,9,13} {1,8,10} {2,3,6,9,12} {1,4,8,13} {2,3,5,6,9,10,11} {1,4,7,8,13} {3,6,11} {1,4,5,7,8,9,13} {2,3,10,12} {1,4,5,7,17} {2,3,6,8,9,13} {4,10} {1,2,3,8,9,13,14} {4,5,6,7,10,11} {1,2,8,9,16} {3,5,10,11} {1,2,4,6,8,14,16} {3,9,10} {1,2,4,5,6,11,13,17} {3,7,8,9,10} {1,2,5,6,11,12,15} {3,4,9,14} {1,2,5,6,7,13,19} {3,4,8,9,14} {1,2,5,6,7,10,12,18} {3,9,11} {1,2,5,7,8,15,16} {3,6,10,12} {1,2,4,5,7,9,13,14,15} {3,6,8,10,12} {1,4,5,9,13,15} {2,3,7,8,10,11,14} {1,4,6,21} {2,3,5,7,9,11,13,15} {1,4,6,8,10,12,16} {2,5,7,9,11,15} {1,3,8,10,19} {2,4,5,6,7,9,12,13,16} {1,3,11,18} {2,4,5,7,9,12,19} {1,11,13} {3,4,5,6,7,9,16,17} {1,2,10,14,15} {4,5,6,8,11,12,13} {1,2,10,21} {3,4,5,7,8,11,12,13,14} {1,2,6,9,10,15,17} {5,7,8,11,12} {1,2,3,4,6,16,18,19} {7,9,10} {1,3,6,11,12,13,15} {4,5,8,18} {1,2,3,6,7,11,14,16,19} {4,5,8,10,17} {2,3,6,12,14,16} {1,4,7,8,10,11,21} {2,3,5,6,9,12,14,20} {1,7,10,18} {2,3,4,6,9,12,14,15,17} {1,5,7,8,10,11,13,18} {2,3,4,6,9,17,23} {1,5,7,10,13,19} {2,3,4,6,8,9,11,12,14,15,17} {1,5,7,10,23} {2,3,4,6,8,11,13,17,19} {1,7,9,20} {3,4,6,8,12,14,18} {1,2,5,7,9,10,11,13,15,20} {4,6,18} {1,3,7,10,12,13,14,15} {2,6,8,11,20} {1,4,5,7,12,18,19} {2,3,6,8,9,11,13,16,17} {1,4,10,23} {2,3,5,7,8,9,11,19,22} {1,4,6,10,13,16,17} {2,8,9,11,18} {3,4,5,6,7,12,17,23} {1,9,10,11,13,14} {2,3,4,6,7,8,12,16,18,21} {1,5,9,10,11,13,19} {2,3,4,6,7,8,12,14,16,17,18} {5,9,10,15} {1,2,3,7,8,13,14,19,21} {5,6,11,12,15} {1,2,3,4,8,9,10,13,14,21,23} {5,6,7,11,12,18} {1,2,4,8,15,19,20} {5,6,7,9,10,11,13,18} {1,2,3,4,12,14,15,16,22} {6,7,9,18} {2,3,5,8,11,12,13,14,16,17} {1,4,6,7,9,10,15,18,21} {2,5,8,11,12,13,14,16} {1,3,6,9,10,20,22} {2,5,7,8,16,23} {1,3,4,6,9,10,11,13,17,18,20} {2,7,8,12,22} {1,3,5,9,10,14,15,17,18} {2,8,12,19} {3,4,5,6,9,11,13,14,15,16,17} {1,2,8,10,12,18,21} {3,4,5,6,7,9,11,16,20,22} {2,10,19} {1,6,7,8,9,11,12,13,14,15,18} {2,3,4,5,10,16,20,23} {1,7,8,15,21} {2,4,5,6,9,10,11,12,13,14,17,22} {1,3,7,8,15,16,23} {2,4,5,10,11,13,14,17,18} {3,6,7,8,16,23} {1,2,4,5,10,11,14,15,17,18,19} {3,6,7,8,9,12,13,16,21} {2,4,5,10,11,19,23} {1,3,6,7,8,9,12,13,14,15,17,22} {2,4,5,19,23} {1,6,7,8,9,16,18,20} {2,3,4,5,11,12,13,14,15,17,21} {6,10,16} {1,2,3,5,8,14,15,18,19,22} {4,6,9,11,12,16,17} {1,2,3,5,8,10,13,14,18,21,23} {7,9,11,16} {3,4,5,6,10,14,15,19,21} {1,7,8,18,20} {2,3,6,10,11,12,13,14,15,16,17} {1,4,8,9,20,23} {2,3,5,6,7,10,11,12,13,14,17,19,22} {1,4,8,9,15,16,23} {2,3,6,7,11,17,19,22} {1,4,8,9,13,14,15,16,18} {2,3,6,7,10,11,12,17,19,22} {1,4,5,13,20,23} {2,3,8,9,10,11,12,14,16,18,19} {1,4,5,6,7,13,15,17,20,23} {2,3,8,10,11,12,14,18,22} {4,5,6,9,13,16,17,19} {1,2,8,11,15,20,21} {3,4,5,6,7,9,10,13,14,16,17,19,22} {1,2,11,12,18,23} {3,4,6,8,9,10,13,15,16,19,20} {1,5,12,17,21} {2,3,6,7,11,13,18,19,22} {1,8,15,21} {2,3,4,7,9,10,14,16,17,19,23} {1,5,8,11,15,18,21} {3,4,7,9,10,12,13,14,19,22} {1,2,5,16,21,23} {3,6,7,8,9,11,13,14,17,18,19} {1,2,4,5,16,20,21,22} {3,6,7,8,9,10,11,12,13,15,17,18,19} {2,4,14,16,21} {1,3,6,7,8,10,11,13,17,18,20,23} {2,4,9,12,15,16,22} {1,3,5,6,13,14,17,21,23} {2,4,7,8,9,11,12,16,18,19,20} {10,13} {2,3,4,5,8,12,15,16,19,20,23} {1,6,7,10,11,13,17,18,21} {2,5,8,9,14,20,23} {1,3,4,6,11,12,13,15,16,17,19,22} {2,5,10,18,23} {1,3,4,7,9,11,12,13,14,17,19,20,21} {2,5,6,8,15,16,18,23} {1,3,4,7,10,12,14,17,19,20,21} {5,8,22} {1,2,7,10,11,13,14,15,21,23} {4,5,6,8,17,20,22} {2,3,7,10,11,12,13,15,16,19,21} {1,9,14,23} {3,4,5,6,8,11,12,13,15,16,19,20,21} {1,2,7,9,10,14,18,22,23} {4,6,13,16,20} {2,3,5,7,9,10,14,17,18,22,23} {4,11,12,13,15,16} {1,3,6,7,8,9,10,14,17,18,19,20,22} {2,4,12,13,15,16,21} {1,6,8,9,14,17,18,22} {2,3,7,10,12,13,19,20,21} {1,4,5,6,11,14,16,17,22,23} {2,3,7,9,10,12,13,15,19,20,21} {1,8,11,18,22} {5,6,7,9,12,13,14,15,16,17,19} {1,2,3,8,10,11,20,21,22,23} {4,6,7,9,13,16,17,18,19} {3,5,8,10,12,14,22,23} {2,7,9,13,15,19,20} {1,3,4,6,8,11,12,14,16,17,21,22,23} {7,9,10,13,15,19} {1,2,3,6,8,14,16,18,21,22,23} {7,9,10,15,20} {2,5,8,11,12,13,17,19,23} {1,3,4,6,9,10,14,15,16,18,20,21,22} {2,11,17,19} {4,5,6,8,10,12,14,15,18,20,23} {3,7,9,11,16,19,21} {2,4,5,6,10,13,18,20,22,23} {3,9,12,14,17,19} {1,4,5,6,11,13,15,16,20,22,23} {2,7,9,10,14,17,19,21} {1,3,5,6,8,11,13,15,16,18,20,22,23} {7,9,10,17,19} {1,4,5,6,11,12,13,15,16,21,22,23} {3,7,8,14,17,18,20} {4,5,9,11,12,15,16,19,21} {2,6,7,8,10,13,14,17,18,20,22} {4,11,16,19} {2,3,5,7,9,14,17,18,20,21,22} {4,6,10,11,12,13,15,19,23} {3,7,9,14,16,17,22} {1,2,4,10,11,12,13,15,19,20,21,23} {3,9,16,17,18} {1,2,6,7,11,12,13,14,15,19,20,21,23} {4,5,9,10,16,17,18,22} {2,3,7,11,12,13,14,15,19,20,23} {1,4,10,18,21,22} {5,6,7,8,13,14,15,16,20,23} {3,4,11,12,18,19,22} {1,6,8,9,10,14,15,16,17,21,23} {4,7,18,22} {5,6,8,12,13,14,16,20,21} {2,7,15,17,23} {4,5,6,9,10,11,14,19,20,21,22} {1,2,16,17,18,23} {3,5,6,8,10,11,13,14,15,19,20,21,22} {4,7,9,12,17,18,23} {3,5,8,11,15,19,20,22} {1,6,9,12,14,16,17,18,23} {5,7,8,10,11,13,15,19,20,21} {1,4,6,9,12,14,16,17,18,22,23} {2,10,19,21} {1,5,7,12,13,15,17,18,20,23} {2,6,8,10,14,16,19,21,22} {1,5,11,13,15,17,20,23} {2,7,9,12,19,21,22} {1,5,8,10,11,13,14,15,16,17,18,20,23} {2,3,12,19,21,22} {4,6,8,9,11,14,15,17,18,20,23} {1,10,12,21,22} {7,8,9,13,14,15,16,18,19} {5,11,17,20} {4,6,8,9,12,14,15,16,19,21,22} {1,7,11,13,18,20,23} {3,5,6,10,14,16,17,19,21,22} {4,7,8,18,20,23} {2,6,11,12,13,14,15,16,17,19,22} {3,4,8,10,18,20,21,23} {1,11,14,19,22} {3,4,7,10,12,13,15,16,17,20,21,23} {2,8,11,14,18,19,22} {5,7,10,12,13,16,17,20,21} {3,6,8,9,11,14,15,18,19,22,23} {5,12,16,21} {1,6,9,10,13,18,20,22,23} {3,11,14,15,16,17,19} {4,6,7,8,9,12,13,18,20,21,22,23} {10,11,14,16,17} {3,4,7,15,18,19,21,22} {5,9,10,11,12,13,14,16,17,20,23} {4,15,22} {2,6,10,11,12,14,18,20,21,23} {5,8,13,15,16,17,22} {4,6,7,9,10,14,18,19,20,21,23} {3,13,17,22} {5,8,10,12,15,16,18,19,21} {3,7,14,22,23} {2,8,10,11,13,15,16,17,19,20,21} {1,5,14,18,22,23} {4,6,15,16,17,19,20} {3,8,10,11,13,21,22,23} {4,9,12,14,15,16,17,18,20} {5,7,8,10,11,13,19,21,22,23} {9,12,15,20} {2,6,10,13,16,17,19,21,23} {5,7,12,14,15,18,20,22} {4,9,13,16,17,19,21} {7,10,11,15,20,22} {4,5,6,13,14,16,17,18,19,21,23} {7,10,12,20,22} {3,6,11,13,15,17,19,21,23} {1,14,20,22} {5,6,8,12,13,15,17,18,19,21,23} {7,10,11,14,16,20,22} {4,5,8,13,15,17,18,19,21,23} {11,12,20} {6,8,9,13,15,19,22,23} {10,11,14,17,20} {1,8,13,16,18,22,23} {6,7,11,14,15,17,19,20,21} {5,12,18,23} {3,8,13,14,15,16,19,21,22} {6,7,11,17,18,20,23} {5,12,15,19,22} {1,7,13,14,18,20,21,23} {3,19,22} {8,9,10,13,14,15,17,18,20,23} {2,11,12,16,19,21,22} {9,13,17,20} {7,10,12,14,15,16,18,19,22} {1,9,20,21,23} {8,10,11,13,14,15,16,17,18,19,22} {4,9,12,20,21,23} {5,14,15,16,17,18,19} {9,10,11,12,13,20,21,23} {5,8,14,15,16,17,18,19,22} {12,13,20} {3,8,15,16,17,18,21,23} {7,11,13,14,19,20,22} {6,12,15,17,18,23} {7,13,16,19,21} {1,9,12,15,17,18,20,22,23} {7,14,19,21} {9,10,12,15,18,20,23} {6,19,21} {11,12,13,14,15,16,20,22} {2,17,18,19,21} {10,11,12,14,16,22,23} {7,9,13,15,17,18,19,20,21} {11,12,16,23} {4,13,14,17,19,20,22} {10,11,16,18,23} {1,12,13,17,19,20,21,22} {6,18,23} {4,10,13,15,16,17,19,20,21,22} {8,14,18,23} {9,10,12,15,16,17,20,21,22} {11,13,14,18,23} {9,10,16,19,20,21} {11,13,14,15,17,18,23} {8,9,12,16,19,20,21,22} {6,17,18,23} {7,11,13,19,20,21,22} {9,14,16,17,18,23} {6,13,19,21,22} {11,12,14,15,16,17,18,20,23} {3,19,21,22} {9,13,14,17,18,20,23} {6,21,22} {8,12,14,17,18,19,20,23} {11,13,15,21,22} {5,14,16,18,19,20,23} {12,15,17,22} {10,13,14,16,19,21,23} {9,15,17,20,22} {10,12,14,16,18,19,21,23} {13,15,22} {7,9,18,19,20,21,23} {13,15,17,22} {8,14,19,20,23} {9,15,16,18,21,22} {12,13,14,17,19,20,23} {11,18,22} {10,15,16,19,20,23} {11,14,18,21,22} {13,17,19,20} {10,11,16,18,21,22,23} {13,19,20} {15,16,17,18,21} {11,13,14,19,20,22,23} {15,16,18,21} {11,12,20,22,23} {13,19,21} {10,15,16,18,20,22,23} {14,17,19,21} {12,16,18,20,23} {13,15,17,19,21,22} {18} {13,15,17,19,22,23} {14,18,20,21} {15,17,23} {12,18,19,21,22} {17,20} {14,16,18,22,23} {17,19,20} {13,18,21,23} {16,17,19,20,22} {15,23} {16,18,21,22} {15,20,23} {17,18,19,21,22} {16,23} {18,20,21} {15,19,22,23} {20} {17,19,22,23} {20,21} {17,22,23} {21} {20,23} {22} {23}
L(24) = 477:  peak size 15 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,9} {3,4,5,6,8} {1,2,7,11} {3,5,8} {1,2,4,6,14} {3,8} {1,2,4,7,14} {3,5,9} {1,2,4,16} {3,5,6,7,8} {2,10} {1,3,6,7,9,11} {2,4,5,8,12} {3,6,7,9} {1,2,4,5,8,11,13} {3,7,9} {1,2,4,10,15} {6,7} {1,2,4,5,10,11,13} {3,6,7,8,9} {2,5,13} {1,4,6,7,8,10,11} {2,3,5,17} {1,4,8,9,12} {2,3,5,7,10,14} {1,9,11} {2,4,5,6,8,10,15} {1,3,7,9,11,12} {2,4,5,10,15} {1,7,9,12} {2,3,4,8,10,11,13} {6,7,9} {1,2,3,4,8,11,14,16} {5,6,7,9,10} {1,2,3,4,13,14,15} {5,6,8,11} {1,2,3,7,9,15,16} {4,5,8,10,11} {1,2,3,6,7,9,12,21} {4,5,14} {2,3,6,7,9,12,15} {1,4,10,16} {2,3,5,6,7,8,9,12,18} {1,4,10,11,13} {5,6,7,8,9,12} {1,2,3,4,10,11,14,18} {5,7,12} {1,2,3,4,11,13,15,16} {5,6,7,8,9,10,12} {1,2,3,4,19,20} {5,6,7,9,14} {1,2,3,4,8,10,11,17,18} {5,6,9,13} {1,2,3,7,8,17,20} {4,5,6,10,12,13} {1,2,3,7,8,9,18,19} {4,5,6,12,15} {2,3,7,9,11,13,14} {4,5,6,8,10,18} {1,2,3,7,9,11,13,14,17} {4,5,6,8,10,12,15} {1,3,7,14,18} {2,4,5,6,10,11,12,19} {1,7,18} {3,4,5,6,8,9,10,12,13,17} {1,2,14,18} {3,4,5,6,9,10,11,12,19} {1,2,8,15,18} {3,5,6,7,11,21} {1,2,4,10,13,14,18} {3,5,6,8,9,11,12,17} {2,4,10,20} {1,3,5,8,9,11,13,15,17} {2,4,6,7,10,12,14,18} {1,3,5,8,13,15,19} {2,4,6,11,12,20} {1,3,5,7,8,9,10,13,14,15,16} {2,4,6,11,23} {1,3,5,7,9,10,15,16,17} {2,4,13,18} {1,6,8,9,12,14,15} {2,3,4,5,7,10,11,13,17,21} {6,8,14} {1,2,3,4,13,15,18,19} {5,6,9,10,17} {1,2,3,11,13,14,22} {4,5,6,7,8,9,10,12,24} {1,2,3,11,19,21} {5,6,7,8,9,10,12,14,15} {1,2,3,11,13,16,21} {4,6,7,8,23} {1,2,3,5,9,15,18,24} {6,7,16} {1,2,3,8,9,10,11,14,18,21} {4,5,6,7,12,15,19} {1,2,3,8,9,10,11,13,14,16,20} {4,6,7,22} {1,2,8,9,10,13,14,15,16} {5,6,7,11,20} {1,2,3,4,8,9,13,14,15,18,21} {5,6,7,10,11,20} {1,3,8,9,12,15,21} {2,6,7,10,11,13,14,16} {1,3,4,5,8,9,15,17,18,19} {2,6,7,12,23} {1,4,5,8,10,11,13,14,15,20} {2,3,6,7,9,12,16,17,19} {1,4,8,10,11,14,15,18} {2,3,5,6,9,22,24} {1,7,8,11,16,18} {2,3,4,5,9,10,12,13,15,17,22} {1,6,7,16,21} {2,4,5,9,10,13,14,15,20} {1,3,16,21} {2,4,6,8,9,10,11,13,15,17,18} {1,3,5,7,12,20,24} {2,4,6,8,9,10,13,14,16,21} {3,5,23} {1,2,4,7,9,10,11,13,17,19,21} {3,5,6,8,12,15,16,18} {2,7,9,13,21} {1,4,5,6,8,10,11,12,14,15,19,20} {2,3,7,9,13,18,21} {4,5,6,8,10,11,14,16,20} {1,3,7,9,21,22} {4,5,6,8,10,11,12,13,14,16,17} {1,2,3,7,9,15,18,19,21} {5,6,8,10,11,12,22} {1,2,3,4,7,9,13,15,16,18,19,20} {5,6,11,14,17} {1,2,4,10,12,13,19,24} {3,5,6,7,8,9,11,14,15,17,22} {1,2,4,12,21,24} {3,6,7,8,9,10,13,15,17,19} {1,4,5,12,14,18,21} {2,3,6,8,9,10,11,13,16,17,23} {4,5,14,20} {1,6,7,9,10,11,13,17,23} {4,5,14,15,16} {1,2,3,7,8,10,12,13,19,20,24} {4,5,6,15,17,18} {1,2,3,7,8,9,10,11,13,16,19,20,22} {4,5,6,12,14,17,18} {2,3,7,8,9,13,21,24} {1,4,5,6,10,15,16,18,23} {2,3,7,8,9,11,12,14,19,24} {5,10,13,16} {1,2,4,7,9,12,14,15,18,19,21} {3,5,6,8,10,11,13,16,17,22} {1,2,4,7,9,14,18,21,24} {3,5,8,11,12,15,16,19} {2,4,7,13,14,17,21} {1,3,5,6,8,9,10,11,12,15,20,22,23} {2,4,13,14,16,18} {1,5,6,7,8,9,10,11,19,23,24} {3,4,14,17,18} {2,7,8,9,10,12,13,16,24} {1,5,18,21} {2,4,6,8,10,11,13,14,16,17,23} {3,5,7,12,15,18,19} {1,2,4,9,10,11,14,17,22,23} {6,7,8,13,16,18} {2,3,5,9,10,11,12,14,17,19,23} {1,4,7,8,13,16,20,22} {2,3,5,6,9,10,11,12,14,15,17,21,23} {4,7,8,16,22} {1,2,3,5,9,11,14,15,17,19,20,21} {4,6,7,10,13,16,24} {1,3,5,8,12,14,18,20,22} {2,4,6,7,9,10,11,13,17,23,24} {1,8,18,19} {2,3,4,5,7,13,14,15,17,23,24} {1,6,8,9,10,11,18,19,22} {2,4,7,13,15,17,23} {1,3,5,6,8,9,10,11,19,21,22,24} {4,7,13,16,18} {1,2,6,8,9,10,11,12,14,15,19,20,24} {3,4,5,7,13,17,21,23} {1,2,6,8,9,12,14,16,18,20,22} {3,5,7,10,21,24} {4,6,8,9,11,12,13,16,18,20} {2,3,7,10,17,19,24} {1,4,5,6,8,11,13,18,20,21,22} {3,7,14,23} {2,5,6,8,9,10,11,13,15,16,17,19,22} {1,3,4,7,12,14,20,21,24} {8,9,10,15,17} {1,4,5,6,7,11,12,19,20,22,23} {2,9,10,13,16,21} {3,4,6,7,8,11,12,14,15,17,18,19,20} {1,5,9,10,13,22,23} {4,6,7,11,12,16,19,20} {1,2,5,8,10,14,21,22,24} {3,4,6,11,12,13,15,17,18,20} {1,2,5,7,8,9,10,16,19,21,22,23} {6,18} {1,2,4,7,9,10,11,14,19,21,23,24} {3,5,6,8,12,13,15,16,17,18,20} {1,4,7,9,10,11,14,19,22,24} {3,5,6,8,13,16,17,20,21} {1,4,10,12,14,15,18,23} {2,3,5,6,7,8,9,11,13,16,17,19,20,22,24} {1,4,10,14,15,18,23} {2,5,6,7,8,11,12,13,16,17,19,20,22} {1,9,10,14,15,24} {2,4,6,7,8,11,12,18,21,22,23} {5,10,13,14,19} {1,2,3,4,7,8,11,15,16,17,18,22,23,24} {5,6,9,10,12,13,14,20,21} {1,2,3,4,7,8,11,17,18,19,22,23,24} {5,10,14,20} {2,3,4,8,11,12,15,16,18,22,24} {1,6,7,13,17,19,23} {4,5,8,10,11,12,14,18,20,21} {1,2,3,6,7,9,13,15,16,19,22,23,24} {4,8,10,11,20,21} {1,3,5,6,7,15,16,18,19,22,24} {2,8,9,11,12,13,21,23} {1,3,4,5,7,10,15,16,17,18,19,22,24} {2,6,13,20,21} {4,5,7,8,9,12,14,15,16,17,18,24} {2,3,6,13,20,21,22} {5,9,10,11,12,14,15,17,19} {1,3,4,6,7,13,16,20,21,22,24} {5,10,11,12,18,19} {1,3,4,6,8,9,17,21,22,23,24} {2,7,11,12,13,15,16,18,19} {1,4,5,10,21,23,24} {2,3,8,9,11,12,14,16,17,18,19,22} {4,5,10,21,23} {1,3,7,8,9,11,12,13,17,18,19,22,24} {2,4,5,10,16,20,21,23} {1,3,8,9,13,14,15,17,18,19,22} {2,12,24} {1,3,4,5,6,14,15,16,18,19,20,21,23} {2,7,9,10,11,12,13,17,22,24} {3,5,8,14,16,20,23} {1,6,7,9,10,11,15,17,19,21,24} {4,8,16,23} {2,3,6,7,10,11,13,14,18,19,20,21,22} {1,5,8,9,12,16,17,23,24} {2,3,4,7,10,11,13,14,15,18,19,20,21,22} {5,8,12,16,23} {1,2,4,7,10,13,18,19,21,22,24} {6,8,9,15,16,23} {1,3,4,7,10,12,13,17,18,19,20,21,22} {2,8,11,14,15,16,24} {1,3,5,6,7,9,10,12,17,18,19,20,21,22,23} {2,8,11,13,14,15,16,24} {1,6,9,10,12,18,19,20,21} {3,4,5,7,11,14,16,22,23,24} {2,6,8,9,10,12,17,18,19,20,21} {1,4,5,7,11,13,14,15,16,22,23,24} {2,12,18,20} {4,5,6,7,9,13,15,16,17,19,22,24} {2,3,8,10,12,14,18,20,21,23} {4,5,6,9,13,16,19,22,24} {3,7,10,12,14,18,20,21} {1,5,11,13,16,22,24} {2,4,6,8,10,12,14,15,17,19,20,21,23} {1,7,13,16,18,24} {2,5,8,9,11,12,14,19,20,22,23} {1,7,16,18,24} {2,9,11,12,13,15,17,19,21} {1,4,5,7,8,10,14,16,18,20,22,23,24} {2,13,17,21} {1,4,7,9,10,12,14,20,22,23,24} {2,8,13,15,16,18,21} {1,6,9,10,12,14,17,19,22,23} {2,5,15,16,18,24} {3,4,7,9,11,13,14,20,21,22,23} {2,6,8,15,16,17,19,24} {1,5,7,9,10,11,13,14,18,20,21,22,23} {4,6,16,17,24} {2,5,7,10,11,13,14,15,18,21,22,23} {4,8,9,12,17,20,24} {5,7,10,11,13,15,18,19,23} {2,3,6,8,12,14,16,20,21,22,24} {5,7,13,15,18,23} {1,2,9,10,12,14,17,19,20,21,24} {3,4,8,13,15,16,18,22,23} {2,5,6,17,20,21,24} {3,7,9,11,12,13,14,15,16,18,22,23} {1,6,17,20,24} {3,5,9,10,12,13,14,15,16,18,19,21,22} {4,6,7,8,17,20,23,24} {1,3,11,12,13,14,16,18,19,21,22} {4,5,6,20,23,24} {3,9,10,11,13,15,16,17,21,22} {1,2,12,14,20,23,24} {3,6,7,11,13,15,16,18,19,21,22} {1,5,8,12,17,20,23,24} {2,7,9,14,15,16,18,21,22} {3,6,17,19,24} {1,5,9,11,13,15,16,18,20,21,23} {3,8,12,14,22,24} {4,5,7,10,11,13,15,16,17,18,20,21,23} {1,8,9,14,19,22,24} {4,10,11,13,15,17,18,23} {1,7,9,12,14,19,20,21,22} {5,6,8,11,13,15,16,18,23,24} {2,7,9,10,12,14,17,19,20,21,22} {8,11,13,24} {3,7,10,12,14,16,17,19,21,22} {6,8,9,11,13,15,18,23,24} {3,4,10,14,19,20,21,22} {5,11,12,13,16,18,24} {1,6,7,9,10,14,15,17,19,20,21,22,23} {2,12,13,16,18,24} {4,5,7,10,11,14,19,20,21,22,23} {3,12,15,17,24} {4,6,8,9,16,19,21,22,23} {7,12,14,24} {2,4,8,11,15,16,18,19,20,21,23} {3,7,13,14,17,22,24} {4,8,10,12,15,16,18,19,20,21} {5,6,13,17,22,23} {2,4,8,11,15,16,18,19,20,21,24} {5,9,12,13,14,17,22,23} {4,10,18,19,21} {5,6,8,11,12,13,14,15,20,22,23,24} {1,9,16,17,18,19,21} {4,8,10,11,13,15,22,23,24} {6,7,9,12,14,16,17,18,19,20,21} {8,10,11,13,22,23} {4,6,9,14,16,18,19,21,24} {7,8,10,15,17,22,23} {1,6,11,12,13,14,16,18,19,20,21,24} {3,8,17,22,23} {4,6,13,14,16,19,21,24} {7,8,9,10,12,15,17,18,20,22,23} {3,5,16,19,21,24} {6,10,12,13,14,15,17,18,20,22} {4,5,7,19,21,23,24} {9,10,11,12,13,14,16,17,18,20,22} {3,6,7,15,19,21,23,24} {4,10,12,14,16,17,18,20,22} {1,5,21,23,24} {7,8,11,12,13,16,17,18,19,20,22} {4,9,14,15,23,24} {10,11,12,16,17,18,20} {2,4,13,14,19,21,22,24} {5,8,12,15,16,17,18,20,23} {6,9,10,11,13,14,19,21,22,24} {4,16,17,23} {1,7,12,14,15,20,21,22,24} {8,9,11,16,17,18,19,23} {4,7,14,15,20,22,24} {6,11,12,18,21,23} {4,9,10,13,14,15,17,19,20,22,24} {7,12,16,18,23} {3,8,9,11,19,20,21,22,24} {10,12,16,23} {6,8,9,11,13,17,18,19,21,22,24} {7,12,14,15,16,20,23} {3,5,11,13,17,18,19,21,22,24} {6,12,15,16,20,23} {5,8,10,11,14,17,18,19,21,22,24} {7,12,15,20,23} {3,6,16,18,19,22,24} {4,12,13,14,15,17,20,21,23} {5,11,22,24} {6,8,14,15,16,17,20,21,23} {4,10,12,18,19,22,24} {8,11,15,21,23} {4,5,14,17,19,20,22,24} {9,15,23} {8,10,11,12,13,16,20,21,22,24} {4,14,15,17,18,19,23} {3,16,20,24} {10,12,13,14,15,17,18,21,22} {7,9,19,20,24} {6,11,12,13,15,16,17,18,21,22,23} {8,10,14,19,20,24} {2,9,16,18,21,22,23} {10,12,13,14,15,19,20,24} {6,9,11,16,17,18,21,22,23} {7,14,15,20,24} {3,12,16,17,18,19,21,23} {5,13,14,15,20,22,24} {8,9,17,19,21,23} {2,15,18,22,24} {4,12,14,16,17,19,20,21,23} {10,15,18,22} {7,9,12,13,14,17,19,20,21,23,24} {11,16,22} {7,9,14,15,18,21,23,24} {8,13,19,20,22} {11,12,14,15,16,23,24} {8,10,13,17,18,19,20,21,22} {12,14,16,24} {8,9,10,17,19,20,21,22,23} {12,13,16,18,24} {5,11,14,17,20,21,22,23} {8,18,24} {9,10,12,15,16,19,20,21,22,23} {8,17,18,24} {10,13,14,15,16,19,20,21,23} {3,17,18,22,24} {9,13,15,20,21,23} {10,12,14,17,19,22,24} {9,13,15,16,18,20,21,23} {8,19,24} {9,14,15,16,21,22,23} {10,12,18,19,20,24} {9,16,17,21,23} {11,12,14,15,18,19,20,22,24} {13,16,17,23} {9,10,18,19,20,21,24} {14,16,22} {7,10,15,19,20,21,23,24} {14,16,17,18,22} {8,11,15,20,21,23,24} {12,17,19,22} {11,14,15,16,20,23,24} {8,18,19,21,22} {12,14,15,16,17,20,23,24} {10,21,22} {12,13,16,17,19,23,24} {14,15,20,22} {8,17,19,21,24} {11,15,16,20,22,23} {12,14,17,18,19,21,24} {15,16,23} {12,13,19,20,21,24} {11,17,18,22,23} {14,16,19,24} {11,13,18,20,21,22,23} {12,19,24} {15,16,17,21,23} {12,14,18,19,20,22,24} {13,17,21,23} {14,16,19,20,24} {13,21,22} {12,16,17,19,20,23,24} {14,18,21,22} {12,15,20,23,24} {16,17,18,19,21,22} {14,20,23} {10,19,21,22,24} {16,18,20,23} {15,19,24} {16,18,20,21,22} {15,24} {14,18,21,22,23} {16,19,24} {14,20,22,23} {17,18,19,21,24} {15,22,23} {17,19,21,24} {18,20,23} {16,19,21,22,24} {18,23} {17,21,24} {18,20,22,23} {21} {17,23,24} {21,22} {18,23,24} {22} {21,24} {23} {24}
L(25) = 537:  peak size 15 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,2,5,9} {3,4,6} {2,7} {1,3,4,6,9} {2,5,7} {1,3,4,11} {2,8} {1,3,4,7,11} {2,5,6,8} {1,4,11} {2,3,6,7,9} {1,4,5,12} {2,3,6,8,9} {1,4,12} {2,5,6,10} {1,3,4,7,14} {2,5,11} {1,3,6,7,8,12} {2,4,5,9,11} {1,3,7,14} {2,4,5,6,8,9,10} {1,7,11} {3,4,6,9,10} {1,12} {2,3,5,6,7,10,13} {1,4,8,9,11} {3,7,10} {1,4,5,6,8,11,12} {2,3,7,15} {1,6,8,9,10} {2,3,4,5,13,14} {1,8,12} {2,3,4,6,10,11,14} {1,5,7,8,9,13} {2,3,4,6,21} {1,7,8,13} {2,3,4,5,9,12,16} {1,6,15} {3,4,5,7,8,9,11,12} {1,2,6,10,18} {3,4,5,7,8,9,16} {1,2,12,15} {3,4,6,7,9,11,13} {1,2,5,14,16} {3,4,6,7,8,9,11,13} {2,5,16} {3,4,6,7,9,10,15} {1,2,5,23} {3,4,6,7,8,9,10,11,12} {1,2,5,13,18} {3,4,7,9,10,14} {1,2,5,6,8,13,20} {3,4,7,18} {1,5,6,8,9,10,12,14} {2,3,4,7,11,13,17} {1,5,6,8,14,15} {2,4,9,10,16} {1,3,5,6,7,8,13,14,17} {2,4,12,15} {1,3,5,6,9,16,18} {2,11,12} {1,3,4,6,8,9,13,23} {2,5,7,11,17} {1,3,4,6,12,14,19} {2,7,9,16} {1,4,5,6,8,10,11,15,17} {2,3,7,9,12,13,14} {1,4,6,15,17} {2,3,7,8,10,11,12,16} {1,4,6,9,14,18} {2,3,5,8,11,12,20} {1,6,7,21} {2,3,4,5,8,9,12,16,20} {1,7,10,11,15} {2,3,5,6,18,19} {1,4,8,9,11,13,16} {2,3,5,6,7,10,12,14,21} {1,4,11,13,16} {2,3,5,6,8,9,14,17,18} {1,4,7,10,11,12,13,15} {2,3,5,6,8,18,22} {1,7,9,11,13,14} {2,3,4,5,6,8,10,12,15,17,19} {1,7,9,13,16} {2,3,4,5,6,11,15,18,19} {1,7,13,16} {3,5,8,9,10,11,19} {1,2,4,6,7,12,13,15,16,17} {3,5,9,10,11,18} {1,2,4,6,8,12,19,23} {3,5,10,11,18} {1,2,7,8,13,14,21} {3,4,5,6,9,10,12,16,20} {1,2,14,21} {3,5,6,7,9,12,13,15,16} {1,2,4,8,10,20,22} {5,6,9,12,16} {1,2,3,4,11,15,18,23} {8,9,12} {1,3,4,5,10,11,13,15,16,19} {2,6,8,9,12,14,17} {1,3,4,5,7,10,11,13,15,16,22} {2,6,12,19} {1,4,5,7,8,9,14,18,22} {2,3,12,13,19} {1,4,5,6,7,8,9,11,16,20,21} {2,3,10,12,15,17} {4,5,7,8,9,11,25} {1,2,3,6,12,15,16,24} {4,5,7,8,9,10,11,17,18} {1,2,12,15,20} {3,4,6,7,9,11,13,14,16,18} {1,2,5,8,10,12,15,17,21} {3,4,6,7,9,13,19,20} {1,2,8,10,15,17,18} {4,5,6,7,14,25} {1,2,3,8,9,10,11,13,15,17,23} {5,6,7,14,19} {1,2,3,10,11,12,13,17,23} {4,8,14,15} {1,3,5,6,7,9,10,12,18,20,22} {2,4,8,13,14,15,16} {1,3,5,6,7,9,10,17,22,23} {2,4,25} {1,3,5,6,10,11,12,13,14,16,23} {2,4,7,8,9,15,17,21} {1,5,10,14,22} {2,3,4,6,7,9,11,13,15,16,19,20} {1,5,8,10,12,14,23} {2,3,6,7,9,13,17,18,19} {1,12,14,15} {2,3,4,7,9,10,11,16,17,18,19} {1,5,6,8,12,13,14,15,21} {2,7,9,10,11,16,19} {1,3,4,5,6,8,12,13,15,18,20,22} {2,7,9,10,25} {3,5,8,11,12,13,14,19} {1,2,4,6,7,10,15,16,17,18,21} {9,11,12} {1,3,4,5,8,10,16,18,20,22} {2,6,7,9,12,15,24} {1,3,4,5,8,11,13,16,17,19,21} {2,6,12,23} {1,4,5,10,11,15,16,17,18} {3,7,9,14,21} {1,4,5,6,8,10,11,13,17,19,25} {2,3,7,9,21,23} {1,4,5,6,8,11,12,13,14,15,16,17,19} {2,3,7,9,10,20,25} {4,5,6,11,13,14,16,18} {1,3,7,8,10,12,15,17,25} {2,4,5,6,9,11,14,18,19,21} {3,7,8,12,16,20} {1,2,5,6,9,13,14,15,17,19,21} {3,4,7,8,10,11,12,16,18,22} {1,2,6,9,13,14,15,17,23} {3,4,7,8,10,12,21,24} {2,5,6,11,17,18,19} {1,3,4,7,8,9,10,12,13,14,16,23,25} {2,5,6,15,18,21} {1,3,4,7,8,9,13,14,19,20,25} {2,6,11,15,22} {1,4,5,7,8,16,17,19,24} {3,9,15,18} {2,4,5,6,7,11,13,14,19,21,22} {1,3,8,9,15,20,23} {2,5,6,7,11,13,14,16,18,21} {1,3,8,17,19,20} {2,5,6,7,10,11,12,13,16,21,22} {1,3,4,8,15,17,19,24} {2,5,6,7,9,10,11,12,13,14,16,20,23} {1,3,15,17,21} {2,4,5,6,7,10,11,12,13,20,22,25} {1,3,8,14,15,16,23} {4,6,7,9,11,12,13,19,22} {1,2,3,5,10,14,15,16,17,20,23} {4,6,11,25} {1,2,5,7,9,12,14,15,16,22,24} {3,4,6,8,10,11,19,20,23} {1,7,9,13,14,15,22} {3,4,5,6,8,10,11,12,16,19,20,25} {2,7,9,18,22} {1,3,4,5,6,11,12,13,15,17,20,21,23} {2,7,8,9,14,16,18,19} {1,3,4,5,6,11,12,20,21,22,23} {7,8,9,14,15,17} {1,2,4,6,10,11,19,20,21,23} {7,8,9,13,14,15,16} {2,3,4,5,6,10,17,18,19,20,25} {1,9,13,24} {2,4,5,6,8,10,11,12,15,16,20,21,23} {1,3,7,9,13,14,17,18,24} {5,6,12,16,20} {2,4,7,8,10,11,13,14,18,19,24} {1,6,9,16,17,22} {2,3,5,7,8,10,11,12,14,18,20,21,23} {1,4,9,13,15,16,25} {3,5,7,11,14,17,18,20} {1,2,4,8,10,12,22,23,25} {3,5,6,9,11,14,15,17,18,21} {1,2,4,8,10,12,13,16,19,22,24} {5,9,11,14,21} {3,4,6,8,10,12,13,15,17,18,19,20} {1,2,5,7,9,11,14,16,21,23,24} {3,4,6,8,10,12,15,18,20,25} {1,5,7,9,11,13,16,23,24} {4,6,10,12,14,15,17,19} {1,2,3,5,7,8,9,11,13,16,18,20,22,23,24} {4,6,10,12,15,17,21} {1,2,3,5,7,9,11,13,18,20,22,23,24} {4,10,12,14,16,17} {1,2,3,7,8,9,15,20,22,23,24} {4,10,14,16,17} {2,3,5,7,8,9,11,12,13,15,20,21,22,23} {1,4,6,10,14,16,17,18,24} {2,3,5,7,8,11,12,13,15,19,20,21,23} {4,9,14,22} {1,2,6,7,10,12,15,18,19,20,25} {4,5,8,11,13,21,24} {1,3,6,7,10,15,17,19,20,25} {2,4,5,8,9,11,12,13,14,16,21,22,23} {1,3,15,17,18,20} {2,5,6,7,9,11,14,16,21,22,23} {1,3,8,12,13,18,20,24} {2,4,5,6,9,11,14,15,16,17,19,21,22} {1,8,10,20,23} {2,3,5,6,7,12,13,16,17,21,22,25} {1,8,10,11,18,19,20} {2,3,4,5,12,16,22,23,25} {6,7,8,9,10,11,13,15,18,19,21} {2,4,20,24} {3,5,6,7,8,10,11,14,15,16,21,22,25} {1,2,4,9,12,13,17,18,19,20,23} {3,5,6,7,10,14,21,22,25} {4,8,11,12,13,17,23} {2,3,5,6,7,9,10,18,20,22,24,25} {1,8,16,17,21} {2,3,4,5,9,11,12,13,18,19,20,23,25} {1,6,7,10,14,17,22,24} {4,5,8,9,11,12,13,16,18,20,23} {6,10,22} {1,2,4,7,12,13,14,15,16,17,18,21,25} {3,5,6,8,9,11,19,20,22,24} {2,7,10,12,15,18,25} {1,3,5,8,9,14,17,19,20,21,23} {4,10,13,24} {1,2,3,6,7,11,14,16,17,19,22,23,25} {4,5,9,10,12,13,18,20,24} {1,2,3,7,8,11,14,15,16,17,19,21,22,23} {5,6,10,18,25} {2,3,4,8,11,13,15,19,20,22,24} {7,9,12,14,17,18} {2,3,5,6,8,10,11,15,16,20,22,24,25} {4,7,9,13,17,19,21} {1,2,5,6,8,10,11,12,15,16,18,20,22,23,24} {3,7,9,13,14,17,19,21} {5,6,8,10,12,15,16,20,24} {2,3,4,11,13,14,17,19,21,25} {1,5,8,9,10,12,15,16,20,22,24} {2,4,6,7,11,13,14,17,18,19,21,23} {3,8,12,20,22} {2,5,6,7,9,10,15,17,18,21,23,24} {1,3,4,11,12,13,14,19,20,22,25} {5,6,7,9,10,15,17,18,21,23} {2,3,11,12,13,14,19,20,24} {1,6,9,10,15,16,23,25} {2,3,4,5,7,8,13,14,17,18,19,20,21,22,24} {6,9,11,12,15,16,23} {1,3,4,5,8,10,14,17,18,20,22,24,25} {6,11,12,13,16,21} {2,4,5,9,10,15,17,18,20,22,23} {6,8,12,19,21} {1,3,5,7,9,10,11,14,15,16,22,23,24,25} {2,4,6,12,17,18,19,20,21} {1,5,7,8,9,10,11,13,14,22,23,24,25} {2,12,18,21} {1,5,6,9,10,15,16,17,19,23,25} {3,7,12,13,14,20,24} {1,2,8,9,15,16,18,19,22,23} {3,4,6,7,10,11,12,13,17,20,21,24,25} {1,8,15,16,18,22} {2,3,7,9,11,13,14,19,20,24,25} {5,6,8,15,16,17,18,22} {1,2,3,9,10,11,12,13,20,21,23,24,25} {4,8,16,17,22} {1,2,5,10,11,13,15,18,19,20,23,24} {7,8,9,12,16,17,25} {1,3,5,13,15,19,20,21,24} {4,6,8,9,10,11,14,16,22,23,25} {1,3,5,7,12,13,15,17,18,19,20,21,24} {4,23} {1,6,8,9,10,16,17,19,20,21,22} {2,3,11,12,13,14,18,24,25} {4,5,8,15,20,21,22} {2,3,6,9,11,13,16,17,18,19,24,25} {7,8,10,20,23} {3,4,5,6,11,12,15,16,18,19,21,22,25} {1,7,10,13,14,17,23,24} {2,3,5,8,12,15,18,19,21,22,25} {1,9,13,16,20,23} {2,4,7,8,11,12,14,15,18,19,21,22,25} {3,5,6,10,13,16,17,20,23,24} {1,9,11,14,18,21,22} {3,5,6,10,12,13,15,19,20,23,25} {9,11,17,18} {1,3,5,6,12,14,15,16,19,20,21,23,24} {2,8,9,10,13,17,18,22,25} {3,4,5,6,11,12,14,15,16,19,20,21,23,24} {1,8,17,18,25} {4,5,9,12,13,14,15,16,19,21,24} {1,2,17,18,20,25} {5,8,9,10,11,12,13,14,15,16,21,22,24} {3,4,6,17,19,23,25} {2,7,8,9,10,11,12,13,15,16,18,20,21,22,24} {3,4,6,14,17,19,23,25} {5,8,9,11,12,16,18,22,24} {2,4,7,10,13,14,20,21,23,25} {3,5,9,11,12,15,16,17,19,22,24} {2,6,7,8,10,13,14,18,20,21,23,25} {3,17,22} {6,8,9,10,11,12,14,15,18,20,21,25} {1,2,4,13,16,17,19,22,23,24} {8,9,10,12,14,15,18,20,21} {2,4,11,13,17,19,22,25} {5,6,12,15,18,20,23} {2,3,7,9,11,13,14,16,17,21,22,24,25} {1,12,15,18,19,20} {3,6,7,10,11,13,16,21,22,23,24} {4,9,15,18,25} {3,7,8,11,12,20,21,22,24} {2,4,9,10,13,14,15,16,17,18,19,23,25} {1,12,20,24} {2,5,8,13,14,15,16,17,19,23,25} {3,7,11,12,21,22,24} {2,4,9,13,15,16,17,19,23,25} {5,14,24} {1,8,9,10,11,15,17,19,21,22,25} {2,6,12,14,16,18,23,24} {1,4,9,10,11,13,15,17,19,20,21,22,25} {3,6,16,23,24} {1,4,9,11,13,14,17,18,19,20,22,25} {3,6,8,16,21,23,24} {2,5,7,13,17,19,20,22,25} {3,6,8,11,14,15,16,18,21,23,24} {1,10,22,25} {5,6,8,11,14,15,18,19,20,21,23} {1,4,9,13,16,17,22,24,25} {5,11,12,15,18,20,21} {1,6,7,10,13,14,16,17,19,23,24,25} {8,9,15,20,21} {1,4,6,7,10,14,17,18,19,22,23,24,25} {9,11,12,13,15,16,20,21} {1,2,5,8,14,18,19,22,23,24,25} {10,12,13,16,17,20} {1,3,4,9,11,14,15,19,21,22,23,24,25} {7,12,13,16,17,18,20} {1,4,9,10,11,14,19,22,23,24,25} {5,16,18,20} {4,9,12,13,14,17,19,22,23} {6,7,15,21,25} {1,3,10,12,16,17,18,19,20,23,24} {6,9,13,14,22,25} {3,4,8,10,11,15,17,18,19,20,21,23,24} {7,9,12,13,16,22,25} {2,6,11,14,19,20,23,24} {3,4,9,15,17,18,21,22,25} {5,8,11,12,13,14,19,20,23,24} {4,7,9,10,15,16,17,18,21,22,25} {5,8,23,24} {3,4,10,13,16,17,20,21,22,25} {5,6,12,14,15,18,19,23,24} {2,10,11,13,17,21,22,25} {4,12,14,16,18,19,23} {3,7,9,10,11,13,15,17,20,21,22,24,25} {5,12,14,18,19,23} {3,4,8,9,15,16,20,21,22,24,25} {6,10,18,19,23} {3,8,11,14,16,17,21,22,25} {4,15,19,23} {6,8,10,11,13,14,16,20,21,24,25} {3,9,15,17,18,22,23} {5,6,7,10,16,19,20,21,24,25} {2,13,14,18,22,23} {4,6,8,9,16,17,19,20,21,24,25} {7,11,13,14,15,18,22,23} {5,6,17,24,25} {8,9,11,12,13,15,16,18,19,20,21,23} {2,4,14,17,22,24,25} {5,10,13,15,16,18,19,20,23} {7,8,9,11,12,14,17,21,22,24,25} {5,10,16,19,20,23} {1,7,12,15,17,18,21,24,25} {8,10,13,14,19,22,23} {3,7,9,12,15,16,17,18,20,21,24,25} {1,13,19,22,23} {3,12,15,16,17,18,20,24} {5,9,10,11,13,14,19,21,22,23,25} {4,15,16,17,18,24} {3,8,10,13,14,19,20,22,23,25} {6,9,15,17,18,21,24} {4,7,11,12,14,16,19,20,22,23,25} {3,15,21,24} {4,12,13,14,17,18,19,22,23} {5,8,20,21,25} {3,9,12,13,14,17,18,19,22,23,24} {5,8,16,20,21,25} {1,10,13,18,22,23,24} {4,11,14,16,17,19,21,25} {6,10,12,13,15,18,22,23,24} {7,9,11,14,16,17,19,20,21,25} {4,13,18,22,23} {7,10,12,15,16,19,20,21,25} {5,9,11,17,18,22,23,24} {4,13,15,16,19,21,25} {3,7,18,22,23,24} {6,12,13,14,15,16,17,19,20,21,25} {8,9,18,22,24} {7,12,14,15,16,17,19,21,25} {2,18,22,23} {5,11,12,14,15,16,17,19,21,24,25} {8,10,13,18,20,22,23} {4,11,12,14,16,17,19,21,24,25} {9,13,15,18,20,23} {7,10,11,16,17,21,24,25} {8,14,15,22,23} {5,6,16,18,21,24,25} {8,12,13,14,17,19,20,22,23} {4,7,18,21,24,25} {11,12,13,14,15,19,20,22,23} {4,8,16,18,21,24,25} {10,11,19,20,23} {7,13,15,16,17,18,22,25} {9,20,21} {10,12,13,14,15,17,18,19,24,25} {4,11,16,20,21,22,23} {5,18,19,25} {11,13,14,15,16,17,20,22,23} {2,18,19,21,24} {9,11,13,14,15,16,17,20,22,23,25} {7,12,18,19,21,24} {6,11,14,17,22,23,25} {9,13,15,16,18,19,21,24} {8,11,12,14,17,20,22,23,25} {3,18,19,21,24} {11,12,13,15,16,22,23,25} {4,14,18,19,20,21,24} {11,13,15,16,23,25} {12,14,18,20,22} {5,6,15,17,19,21,23,24,25} {9,18,20,22} {7,10,17,19,21,23,24} {14,18,20} {11,12,13,15,17,22,24,25} {7,18,19,20,23} {11,12,14,17,21,22,25} {9,13,15,16,18,19,20,23,24} {14,21} {9,10,11,18,19,20,22,24,25} {13,14,17,21,23} {9,10,12,19,20,22,24,25} {15,17,21} {9,11,14,19,22,24,25} {12,15,21,23} {9,11,14,16,19,20,22,24,25} {10,17,18,21,23} {1,15,20,22,24,25} {13,14,17,18,19,21,23} {9,12,15,16,20,22,24,25} {18} {13,15,16,19,20,21,23} {3,17,18,22,24,25} {13,16,19,20,23} {11,12,14,17,18,21,22,24,25} {15,16,19,23} {11,14,17,18,21,22,25} {12,19,24} {9,14,16,18,20,22,23,25} {11,17,19,21,24} {9,12,18,20,22,23,25} {10,19,21,24} {13,14,15,18,22,23,25} {12,17,19,21,24} {10,15,16,18,20,22,23,25} {11,21,24} {14,16,17,19,20,22,23} {5,21,24,25} {15,17,19,20,23} {11,16,18,21,22,25} {14,15,17,19,20,23,24} {13,25} {15,16,19,20,21,24} {8,18,22,23,25} {13,19,21,24} {12,16,17,20,22,23,25} {13,21,24} {15,18,19,20,25} {13,16,17,21,22,23,24} {15,18,20,25} {12,17,22,23,24} {14,20,25} {15,17,18,19,22,23,24} {13,20,21,25} {17,18,19,22,23} {13,16,20,21,24,25} {17,23} {12,19,21,24,25} {16,20,22,23} {15,21,25} {18,19,20,22,23} {17,24} {18,19,20,21,25} {15,23,24} {16,20,22,25} {17,19,21,23,24} {16,22,25} {18,20,23,24} {17,22,25} {19,20,21,23,24} {18,25} {20,22,23} {17,21,24,25} {22} {19,21,24,25} {22,23} {19,24,25} {23} {22,25} {24} {25}


Bubbler
Posts: 28
Joined: Sat Feb 23, 2013 6:12 am

### Re: Ex-Increasing Set Sequence

Excellent work, thundre. I've been busy for some time recently and couldn't work on this problem.

The conjecture quantheory suggested seems to be false from the sample sequences (seeing {2,8} instead of {5} in L(10) and many others), and also my own thought of symmetry (formally: {x1,...,xi} implies {n+1-x1,...,n+1-xi}) is rejected. Now seems "DFS after culling" is the best strategy for attacking the problem.
thundre
Posts: 356
Joined: Sun Mar 27, 2011 10:01 am

### Re: Ex-Increasing Set Sequence

Bubbler wrote:Excellent work, thundre. I've been busy for some time recently and couldn't work on this problem.

The conjecture quantheory suggested seems to be false from the sample sequences (seeing {2,8} instead of {5} in L(10) and many others), and also my own thought of symmetry (formally: {x1,...,xi} implies {n+1-x1,...,n+1-xi}) is rejected. Now seems "DFS after culling" is the best strategy for attacking the problem.
{5} could be substituted for {2,8} in the L(10) case, since it does not intersect with the predecessor or successor sets. The choice is a little bit random because I used HashSet. If I changed it to a TreeSet sorted by mask, it would always process the singleton set before any others with the same average. I'll try that when I have time.

Code: Select all

L(1) = 1:  peak size 1 -- {1}
L(2) = 2:  peak size 1 -- {1} {2}
L(3) = 3:  peak size 1 -- {1} {2} {3}
L(4) = 5:  peak size 2 -- {1} {2} {1,4} {3} {4}
L(5) = 7:  peak size 2 -- {1} {2} {1,4} {3} {2,5} {4} {5}
L(6) = 10:  peak size 3 -- {1} {2} {1,4} {3} {1,4,5} {2,3,6} {4} {3,6} {5} {6}
L(7) = 15:  peak size 4 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,6,7} {4,5} {1,6,7} {5} {4,7} {6} {7}
L(8) = 21:  peak size 4 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {3,6} {2,5,7} {1,4,6,8} {5} {1,7,8} {5,6} {2,7,8} {6} {5,8} {7} {8}
L(9) = 29:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {2,3,5,7} {1,4,8} {3,6} {1,2,4,7,9} {3,5,6} {1,2,7,9} {4,6} {1,5,7,8} {3,4,9} {1,5,6,7,8} {2,9} {3,6,8} {2,5,7,9} {6} {2,8,9} {6,7} {3,8,9} {7} {6,9} {8} {9}
L(10) = 39:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {2,8} {1,3,5,7,10} {2,4,6,9} {1,5,10} {2,4,6,7,8} {1,10} {2,5,6,7,8} {3,4,10} {2,6,7,8} {1,4,5,9,10} {6} {3,5,8,9} {2,7,10} {4,5,6,8,9} {3,10} {5,7,8} {2,6,9,10} {7} {4,6,9,10} {7,8} {4,9,10} {8} {7,10} {9} {10}
L(11) = 49:  peak size 5 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,4,11} {5} {1,3,6,7,9} {2,4,5,10} {1,6,9} {3,4,5,7,8} {2,9} {1,3,6,7,11} {4,5,8} {2,3,7,11} {1,4,6,8,10} {3,9} {2,5,6,8,10} {3,4,7,11} {1,8,10} {2,4,6,9,11} {3,10} {4,5,6,7,11} {2,8,10} {3,6,7,11} {2,5,8,9,10} {7} {3,6,9,11} {5,7,10} {3,6,8,9,11} {5,10} {6,8,9} {3,7,10,11} {8} {4,10,11} {8,9} {5,10,11} {9} {8,11} {10} {11}
L(12) = 63:  peak size 7 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,5,9} {1,4,10} {2,3,6,7,8} {1,4,5,11} {2,6,8} {1,3,4,7,12} {2,9} {1,4,6,7,10} {3,5,9} {2,4,6,11} {1,3,5,8,12} {2,10} {3,4,5,6,7,12} {1,2,8,9,11} {3,5,7,10} {1,2,4,6,8,11,12} {3,7,9} {1,4,6,10,11} {5,8} {2,3,4,6,9,10,12} {1,8,11} {2,3,5,6,9,10,12} {1,7,8,11} {4,5,6,9,10} {1,2,7,8,11,12} {4,10} {3,5,7,8,9,11} {1,6,10,12} {3,8,11} {2,6,7,10,12} {4,11} {5,6,8,9,10} {4,7,12} {2,8,10,11} {5,6,7,9,12} {8} {5,7,10,11} {4,9,12} {6,7,8,10,11} {5,12} {6,9,11} {5,8,10,12} {9} {6,8,11,12} {9,10} {6,11,12} {10} {9,12} {11} {12}
L(13) = 79:  peak size 7 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,9} {2,3,6,7,8} {1,4,5,11} {3,6,7} {1,2,4,8,12} {5,6} {2,3,4,7,12} {1,6,10} {2,4,8,9} {1,3,6,7,12} {2,5,8,9} {1,3,4,6,10,13} {2,5,7,8,9} {3,4,6,12} {1,2,5,7,8,10,11} {3,4,12} {2,6,7,8,9} {3,10} {2,4,5,6,7,9,13} {1,3,8,10,11} {4,7,9} {2,3,5,6,8,11,12} {1,7,9,10} {2,5,6,8,13} {1,3,7,9,10,11} {4,5,6,13} {2,3,8,9,10,11} {1,4,6,12,13} {3,7,9,10} {1,2,4,8,11,12,13} {3,9,10} {4,6,7,8,12} {1,3,5,9,10,11,13} {4,6,8,12} {2,3,5,9,10,11,13} {4,7,12} {3,5,6,8,9,10,13} {1,7,11,12} {4,5,8,9,13} {1,6,7,10,11,12} {3,13} {4,6,8,9,10,12} {2,7,11,13} {3,10,12} {5,7,8,9,13} {1,10,11,12} {6,7,8,9,13} {4,10,12} {5,8,9,13} {4,7,10,11,12} {9} {4,8,10,11,13} {7,9,12} {5,8,10,11,13} {7,12} {6,10,13} {7,9,11,12} {10} {6,12,13} {10,11} {7,12,13} {11} {10,13} {12} {13}
L(14) = 99:  peak size 8 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,2,5,9} {3,4,6} {2,7} {1,3,4,6,9} {2,5,7} {1,3,4,11} {2,8} {1,3,5,7,10} {2,4,6,9} {1,3,12} {2,4,5,6,10} {3,8} {1,2,5,9,11} {3,4,10} {1,2,8,12} {3,4,5,6,11} {1,2,7,8,12} {3,4,5,6,9,10} {1,2,7,8,13} {3,5,6,11} {1,2,4,7,8,9,13} {3,5,11} {2,6,7,8,9} {1,4,10,11} {2,3,5,6,8,9,13} {1,4,7,10,11} {3,8,9} {1,2,4,5,10,11,14} {3,7,8,9} {1,4,6,10,13} {2,3,5,8,11,12} {1,4,9,14} {3,5,6,7,8,10,11} {1,2,4,9,13,14} {3,5,7,10,11} {2,4,9,14} {1,3,6,8,10,11,12} {4,5,13} {1,2,6,7,8,10,11,14} {3,4,5,12,13} {1,2,6,8,10,11,14} {3,5,9,13} {2,4,6,7,8,12,14} {3,5,9,10,11} {1,2,4,7,8,12,13,14} {3,9,11} {1,4,5,7,10,13,14} {2,6,11,12} {4,5,8,9,13} {1,2,7,11,12,14} {4,5,6,8,9,10,13} {2,14} {1,5,8,9,10,11,13} {2,6,7,12,14} {3,8,9,13} {4,5,6,7,10,12,14} {3,9,13} {4,5,7,12,14} {2,6,8,9,10,11,13} {3,14} {1,6,11,12,13} {2,10,14} {4,6,7,8,11,12,13} {2,9,10,14} {6,7,8,11,12} {2,5,9,10,13,14} {4,11,12} {2,7,9,10,13,14} {6,8,11,12} {5,9,14} {6,8,10,11,12} {2,9,13,14} {7,8,10,11,12} {6,9,14} {7,8,11,13} {4,9,10,12,14} {7,13} {6,9,12,14} {8,10,13} {6,9,11,12,14} {8,13} {7,11,14} {8,10,12,13} {11} {8,10,13,14} {11,12} {8,13,14} {12} {11,14} {13} {14}
L(15) = 121:  peak size 9 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,3,5,8} {2,4,7} {3,6} {1,2,4,7,9} {3,5,6} {1,2,4,7,10} {5} {1,2,6,7,10} {3,4,5,9} {1,7,8} {2,3,4,5,13} {1,6,7,8} {2,3,4,9,10} {1,5,11} {2,3,4,14} {1,5,6,7,10} {2,3,8,11} {1,4,5,6,9,12} {2,3,7,8,11} {4,5,6,10} {1,2,3,7,8,11,12} {4,5,10} {1,3,6,9,13} {2,5,7,12} {1,3,4,8,9,10,11} {2,5,6,7,13} {1,4,15} {2,3,5,6,9,10,12} {1,4,8,14} {2,3,7,10,12} {1,4,6,8,9,13} {3,11} {1,2,4,6,10,12,15} {3,5,7,8,9,11} {1,4,6,10,15} {2,5,9,13} {3,4,6,7,8,11,12} {2,5,15} {1,3,4,7,8,10,12,14} {2,5,6,11,13} {1,3,4,8,10,12,14} {6,9} {1,4,5,8,10,11,14} {3,6,7,9,13} {1,2,4,5,8,12,14,15} {3,7,13} {1,2,4,6,12,14,15} {3,7,10,11} {1,2,5,6,8,9,12,13,14} {3,4,7,10,15} {1,5,6,9,12,14} {2,4,7,8,10,11,13} {3,5,6,12,14} {1,2,7,9,10,13,15} {3,4,5,11,12,14} {1,8,9,10,13} {2,3,4,6,7,11,12,14,15} {1,9,10,13} {2,3,6,7,11,14,15} {4,9,12} {1,3,5,6,10,13,14,15} {4,7,8,11,12} {2,5,6,9,10,13,14} {3,8,11,12} {4,5,6,7,10,13,15} {1,8,9,11,14} {3,4,5,7,10,12,13,15} {1,11,14} {3,5,6,7,12,13,15} {2,9,10,14} {6,7,8,11,12} {1,4,9,10,14,15} {5,6,7,8,11,12,13} {3,10,14} {1,7,8,11,13,15} {5,6,9,12,14} {1,10,11,15} {3,6,8,9,12,13,14} {2,11,15} {3,9,10,12,13} {5,6,7,8,11,14,15} {3,10,12,13} {4,6,8,9,11,14,15} {7,10,12} {5,6,8,9,11,14,15} {4,10,12,13} {5,7,8,14,15} {4,9,10,11,12,13} {7,8,15} {6,9,10,12,14} {2,11,13,15} {8,9,14} {6,10,11,12,13} {5,8,14,15} {7,10,11,12,13} {8,9,15} {6,11,12,14} {7,9,10,13,15} {11} {6,9,12,14,15} {10,11,13} {7,9,12,14,15} {10,13} {9,12,14} {8,11,13,15} {12} {8,14,15} {12,13} {9,14,15} {13} {12,15} {14} {15}
L(16) = 145:  peak size 9 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,9} {3,4,5,6,8} {1,2,7,11} {3,4,9} {1,2,5,6,13} {3,8} {1,4,6,7,10} {2,3,12} {4,5,6,8} {1,2,3,10,13} {4,5,9} {2,3,6,7,8,11} {1,4,5,9,12} {2,6,7,10} {1,3,4,5,8,11,12} {2,7,10} {1,3,4,5,6,8,9,15} {2,11} {1,3,4,6,9,10,13} {2,5,7,8,11} {3,4,13} {1,2,6,8,9,10,11} {3,4,7,13} {1,2,5,12,14} {3,4,7,8,9,10} {1,5,6,16} {2,3,4,7,8,11,15} {1,5,6,9,10,12} {2,3,7,8,16} {1,4,9,15} {2,3,6,7,10,11,12} {1,8,13} {2,3,5,6,7,10,12,14} {1,4,8,9,15} {2,3,5,6,11,12,13} {1,4,10,15} {2,3,6,8,9,11,14} {1,4,7,10,16} {2,3,5,6,9,11,12,13} {1,4,7,8,10,16} {2,3,5,6,11,13,14} {1,4,10,16} {2,3,5,6,7,9,11,12,15} {1,4,8,10,16} {2,5,7,9,11,13} {1,3,4,8,10,14,15} {2,7,11,12} {1,3,6,8,9,10,13,15} {2,4,5,7,11,12,16} {3,6,8,9,10,13} {2,4,7,12,16} {1,3,5,8,9,10,11,13,14} {4,6,7,16} {1,5,8,9,10,12,13} {4,6,15} {2,3,5,7,9,11,14,16} {1,6,8,12,15} {2,4,7,9,10,11,16} {5,12} {2,4,6,8,9,15,16} {3,5,10,12,13} {1,2,4,6,11,14,15,16} {5,9,12} {1,4,7,8,10,15,16} {3,9,11,12} {1,2,4,6,8,13,14,15,16} {5,7,9,11,12} {1,3,6,13,14,16} {4,5,8,9,10,11,15} {1,2,6,7,12,13,14,16} {8,10} {1,5,6,9,12,15,16} {3,4,10,11,13,14} {2,8,9,12,15} {3,4,5,7,10,11,13,14,16} {1,9,12,15} {2,4,8,10,11,14,16} {1,12,15} {2,4,7,9,10,13,14,16} {3,6,11,12,15} {1,5,7,10,13,14,16} {4,15} {3,7,8,9,11,13,16} {2,5,12,14,15} {4,6,8,9,10,11,13,16} {2,12,15} {4,6,7,10,11,14,16} {3,8,13,15} {1,6,12,14,16} {5,7,9,10,13,15} {2,6,8,11,12,14,16} {10} {3,6,8,11,13,14,16} {5,9,10,12,15} {6,8,11,16} {1,7,10,12,13,14,15} {4,11,16} {6,8,10,13,15} {4,7,9,11,12,14,16} {8,13} {5,6,12,14,16} {7,10,15} {5,6,9,12,13,14,16} {7,10,11,15} {5,8,12,13,16} {6,9,10,11,14,15} {5,12,16} {8,9,11,13,15} {3,12,14,16} {8,11,15} {6,9,12,14,16} {8,15} {5,10,13,14,16} {8,12,15} {7,11,13,16} {8,10,12,14,15} {7,13,16} {8,12,14,15} {10,11,16} {8,12,13,14,15} {9,16} {10,13,15} {9,12,14,16} {13} {9,15,16} {13,14} {10,15,16} {14} {13,16} {15} {16}
L(17) = 171:  peak size 10 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,4,5,8} {1,3,11} {2,4,5,6,9} {1,3,7,10} {2,5,9} {1,3,4,6,13} {2,5,7,8} {1,3,4,6,14} {2,7,8} {1,3,9,10} {2,4,5,7,11} {1,3,8,12} {2,4,5,6,9,11} {1,3,7,8,12} {4,5,6,10} {1,2,3,7,8,11,12} {4,5,10} {1,3,6,9,13} {2,4,8,12} {1,3,5,6,7,11,13} {2,4,8,9,10} {1,6,13} {2,3,4,7,9,10,12} {1,5,6,15} {2,4,7,8,13} {1,3,5,6,10,16} {2,12} {3,4,5,6,9,10,13} {1,2,7,8,11,14} {3,4,5,9,15} {1,2,12,14} {3,4,5,6,7,9,17} {2,8,12} {1,3,4,6,7,10,11,17} {2,5,8,9,13} {1,3,4,6,10,12,16} {2,5,9,14} {3,4,6,7,10,11,12} {1,2,8,13,14} {3,4,5,7,9,10,11,12} {2,6,15} {1,4,5,7,11,12,14} {2,3,10,16} {1,4,5,6,7,9,11,13,14} {2,3,8,10,16} {4,5,6,7,11,14} {1,2,3,8,9,15,17} {6,10} {3,4,5,7,8,11,12,15} {1,2,6,9,10,13,16} {3,4,5,8,12,17} {1,2,10,13,15} {3,4,6,7,8,9,11,12,14} {1,2,13,17} {3,4,7,8,10,12,14} {1,2,5,11,15,16} {3,4,7,8,9,10,12,14} {2,5,6,13,16} {3,4,7,9,10,11,15} {2,5,13,14} {1,4,6,7,8,9,10,15,17} {2,3,5,11,12,13,14} {1,4,6,15,17} {3,5,7,8,9,10,11,16} {2,4,6,12,13,15} {1,3,5,7,8,9,10,11,16,17} {2,6,13,14} {1,4,5,7,9,10,12,15,16} {2,3,8,14,17} {1,4,9,11,12,16} {2,6,7,8,10,14,15} {1,3,4,5,12,13,16,17} {2,8,11,15} {1,3,7,9,10,12,14,17} {2,4,6,8,13,15,16} {1,7,10,11,12,14} {3,4,9,13,17} {1,2,5,8,10,12,14,15,16} {4,7,9,17} {2,5,6,10,13,14,15} {3,9,16} {2,5,7,10,11,12,13,15} {1,4,9,16,17} {5,7,8,10,11,12,13} {1,3,4,6,9,14,15,16,17} {7,8,10,13} {2,3,6,9,14,16,17} {1,7,12,13,15} {2,3,4,10,11,14,16,17} {5,9,15} {2,3,7,11,12,16,17} {5,6,13,15} {3,4,7,8,9,10,14,16,17} {5,6,11,12,15} {1,4,10,13,14,17} {5,6,7,9,11,15,16} {2,3,8,10,12,13,14,17} {6,7,11,16} {2,8,9,10,12,13,17} {1,4,11,14,15,16} {6,7,9,12,17} {1,4,8,10,11,13,14,15,16} {3,9,12,17} {1,2,11,13,14,15,16} {4,10,17} {3,6,7,9,13,14,15,16} {4,8,11,12,17} {3,6,9,10,14,15,16} {8,13} {1,9,10,11,12,15,16} {3,6,13,14,17} {4,8,9,10,11,12,15,16} {5,13,14} {1,6,8,12,15,16,17} {9,10,11,13} {3,6,12,16,17} {7,8,10,11,14,15} {3,6,9,12,13,16,17} {7,15} {8,9,10,11,12,17} {6,7,13,14,16} {2,11,15,17} {5,9,10,12,13,14,16} {8,11,15} {5,7,12,16,17} {8,9,10,11,13,14,15} {7,16} {5,10,12,14,17} {8,11,16} {6,7,10,13,14,15,17} {8,11,12,16} {7,10,13,14,15} {6,9,11,12,16,17} {8,13,15} {9,10,12,14,16} {8,11,13,17} {9,12,16} {5,11,14,15,17} {9,16} {7,10,14,15,17} {9,13,16} {7,12,15,17} {10,11,13,14,16} {8,12,15,17} {10,13,14,16} {11,12,17} {9,13,14,15,16} {10,17} {11,14,16} {10,13,15,17} {14} {11,13,16,17} {14,15} {11,16,17} {15} {14,17} {16} {17}
L(18) = 203:  peak size 11 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {1,2,5,9} {3,4,6} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,6,8} {2,3,5,7,9} {1,4,6,10} {2,5,9} {1,3,4,7,12} {5,6} {1,2,4,7,14} {3,6,8} {2,4,5,12} {1,3,6,8,11} {4,5,9} {1,2,3,8,11,12} {4,5,6,7,9} {1,2,10,12} {3,4,5,6,7,8,11} {1,2,16} {4,5,6,7,10} {1,2,8,15} {3,4,5,6,7,10,11} {1,2,8,9,13} {4,5,11} {1,3,6,7,8,10,12} {2,4,5,16} {3,6,7,8,10} {1,2,5,9,11,13} {3,6,12} {1,2,5,7,9,11,15} {3,4,6,8,10,12} {1,2,9,11,13} {3,4,7,15} {1,2,5,6,8,13,16} {3,9,10} {1,2,4,6,8,11,12,15} {3,5,7,9,13} {1,2,4,8,10,12,15} {6,9} {1,2,3,10,11,12,14} {4,6,7,8,13} {1,2,3,5,9,10,14,17} {4,7,12} {1,2,5,6,11,13,16} {3,7,9,12} {1,2,4,5,8,10,11,13,16} {3,6,7,9,14} {1,5,8,10,11,12} {2,3,4,6,9,13,18} {1,7,16} {3,4,5,6,8,12,13,14} {1,2,7,9,10,11,17} {3,4,5,8,14,15} {1,6,9,12,13} {2,3,4,5,7,10,11,14,18} {1,8,9,15} {2,3,5,6,11,13,18} {1,8,16} {4,5,6,7,9,10,11,15} {2,3,8,13,16} {4,5,6,7,9,11,17} {1,16} {2,3,4,6,7,9,14,15,17} {1,5,8,10,11,12,13} {3,4,7,14,15} {2,5,6,8,9,10,13,16} {1,3,4,12,14,18} {2,5,6,8,9,15,16} {1,7,13,14} {2,3,4,5,8,11,12,16,18} {1,6,10,13,14} {2,4,5,8,16,18} {3,6,7,9,11,12,14} {1,2,4,5,8,16,17,18} {3,10,14} {1,4,6,7,8,11,12,15,18} {2,3,5,9,10,13,14,17} {1,4,7,8,11,15,18} {2,5,6,9,16,17} {1,3,4,7,8,10,11,12,13,14,18} {2,5,6,16,17} {3,4,7,8,9,10,11,13,18} {2,6,14,15} {1,3,5,9,12,17,18} {2,4,6,7,8,10,11,14,15,16} {1,3,5,12,17,18} {2,4,8,9,10,13,14,15} {1,6,7,16,17} {2,3,9,10,11,13,18} {1,4,5,6,7,14,15,16,17} {2,11,12,13} {1,4,5,8,9,10,14,17,18} {2,3,6,12,13,15,16} {1,7,9,14,17} {2,3,4,8,11,15,16,18} {6,9,14} {1,4,5,7,8,10,12,15,17,18} {2,3,9,11,13,14,16} {5,7,10,17} {1,4,6,8,11,13,14,15,16} {3,7,9,12,18} {1,2,5,6,8,11,13,14,15,16,17} {3,7,9,10,12,18} {2,4,6,11,13,16,17} {1,7,8,9,10,12,14,18} {2,4,5,6,11,13,15,16,17} {10} {1,5,7,8,11,12,13,16,18} {2,4,9,10,14,15,17} {3,5,7,12,16,18} {1,4,14,15,17} {5,6,8,9,10,11,12,13,18} {2,7,15,17} {5,6,8,9,10,16,18} {2,4,11,13,15,17} {1,5,8,9,12,14,16,18} {2,7,11,15,17} {4,8,9,10,12,14,16} {2,5,6,7,11,13,15,17,18} {4,8,14,16} {1,3,9,11,15,17,18} {4,6,13,14,16} {3,5,8,9,10,15,17,18} {7,12,13} {2,6,9,11,14,15,18} {8,10,12,13} {2,4,7,9,11,14,15,17,18} {5,8,12,13,16} {4,9,10,11,14,17} {3,6,8,12,13,16,18} {4,7,9,10,11,14,15,17} {6,8,12,18} {3,5,11,13,14,15,17} {6,7,8,12,16,18} {5,9,11,14,17} {3,6,8,10,12,13,15,16,18} {5,9,14,17} {3,7,11,12,13,15,18} {4,14,16} {3,7,8,11,12,15,17,18} {4,10,13,14,16} {3,6,9,12,15,17,18} {5,11,14,16} {6,8,9,12,13,15,18} {7,10,11,14,16} {3,6,9,12,13,15,17,18} {8,11,16} {1,9,12,13,14,15,18} {4,10,16,17} {1,12,13,15,18} {8,9,10,11,16,17} {5,6,12,13,14,15,18} {9,10,17} {6,8,11,13,14,15,18} {7,9,12,16,17} {3,13,15,18} {6,10,11,12,14,16,17} {9,13,15} {6,10,11,17,18} {8,9,12,13,14,15,16} {4,11,17,18} {7,12,13,15,16} {10,11,17} {7,8,12,13,15,16,18} {9,11,14,17} {8,10,12,16,18} {7,11,13,14,15,17} {6,12,16,18} {7,13,14,15,17} {8,11,16,18} {9,14,17} {11,12,13,15,16} {9,18} {10,11,14,16,17} {8,15,18} {11,13,14,17} {8,12,15,16,18} {14} {9,13,15,16,18} {12,14,17} {10,13,15,16,18} {12,17} {11,15,18} {12,14,16,17} {15} {11,17,18} {15,16} {12,17,18} {16} {15,18} {17} {18}
L(19) = 237:  peak size 11 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,6,9} {5} {1,3,6,7,9} {2,4,5,10} {3,6,7} {1,2,4,5,15} {3,8} {1,2,4,5,16} {3,6,8} {1,5,7,10} {2,4,6,8,9} {1,3,14} {2,4,6,7,8,10} {1,3,5,9,13} {2,6,7,10} {1,3,4,5,8,9,14} {2,6,11} {3,4,5,7,13} {1,2,8,15} {3,4,5,6,7,10,11} {1,2,8,9,13} {4,5,11} {2,3,6,7,8,9,12} {1,4,5,17} {2,3,8,9,12} {1,4,5,6,7,18} {2,3,9,14} {1,4,5,6,10,11,13} {2,3,7,8,9,14} {1,4,5,10,16} {3,7,8,11} {1,2,4,9,10,12,13} {3,5,14} {1,2,4,6,10,11,12,13} {3,5,7,8,14} {1,2,4,6,10,12,17} {3,5,7,15} {1,4,6,8,10,11,13} {2,5,7,9,15} {1,3,4,6,8,12,13,14} {2,10,11} {1,4,5,6,7,13,18} {3,8,9,11} {1,2,4,6,7,10,12,13,15} {3,5,8,9,14} {1,4,6,11,12,13} {2,3,5,8,9,10,18} {4,7,13} {1,2,6,9,10,11,12,14} {3,4,5,7,8,13,17} {1,2,6,11,14,15} {3,5,7,8,18} {1,2,4,6,9,11,12,14,15} {3,5,7,18} {1,2,6,10,11,12,16} {3,5,7,8,9,18} {1,2,4,6,10,12,15,17} {3,5,7,9,18} {1,2,4,8,10,15,19} {3,5,6,7,13,17} {1,2,4,8,10,11,12,14,15} {3,5,6,7,9,13,17} {2,4,8,11,18} {1,3,5,6,10,13,15,16} {2,4,9,11,12,14} {1,5,6,8,10,15,16} {3,7,11,14} {1,2,4,5,9,10,12,17,19} {3,7,8,11,15} {1,2,4,13,14,19} {3,5,6,7,9,15,17} {1,2,4,8,10,13,14,19} {5,6,16} {2,3,4,7,8,12,14,15,17} {1,5,6,9,10,11,13,18} {2,4,7,8,12,14,17} {1,6,9,10,11,18} {2,3,4,5,7,8,12,13,14,16,17} {1,6,9,11,19} {3,4,5,7,8,10,12,16,18} {1,9,13,14} {2,4,6,8,11,16,18} {1,3,5,7,9,10,12,13,14,19} {2,8,18} {1,4,6,9,11,12,15,17} {3,5,7,14,18} {1,6,8,9,12,13,17} {2,3,4,5,7,14,15,16,19} {1,8,12,17} {2,3,4,5,6,10,11,14,15,16,19} {1,7,8,9,12,13,17} {3,5,11,14,15} {1,6,8,9,10,12,13,18} {2,11,16} {1,3,4,5,7,13,14,15,17,18} {2,8,9,10,11,12,16} {1,3,4,5,6,7,13,14,17,18,19} {2,9,12,16} {4,5,6,7,8,11,13,15,19} {2,3,12,14,18} {1,5,6,7,8,9,11,13,15,16,17} {2,3,10,12,14,18} {1,5,7,9,11,17,19} {2,3,4,12,13,14,15,16} {1,6,7,8,9,10,11,18,19} {3,17} {2,4,5,7,12,13,14,15,19} {3,6,8,9,10,11,16,18} {1,5,7,12,14,15,17} {4,6,8,11,13,19} {1,2,3,5,9,12,14,15,16,17,18} {6,7,8,11,19} {2,3,5,9,10,12,16,17,18} {6,8,13,14} {1,2,3,7,9,10,11,16,17,18,19} {4,6,8,12,13,14,15} {1,2,3,7,9,11,16,17,18,19} {6,12,13} {1,4,7,10,11,14,17,19} {3,6,9,16,18} {1,4,7,12,13,17,19} {2,3,5,10,11,14,15,16,18} {9,12} {2,4,5,7,13,14,15,17,18} {1,8,9,10,11,16,19} {5,6,12,13,17} {3,7,8,9,11,14,15,18} {5,10,17} {1,2,6,8,9,13,15,16,18,19} {4,7,10,11,12,14,17} {3,9,15,16} {4,5,7,8,10,13,14,17,19} {2,3,15,16,18} {1,5,8,9,10,11,12,13,14,17,19} {3,6,7,15,16,18} {4,8,9,10,12,14,19} {1,5,7,11,13,15,17,18} {4,6,8,9,10,12,14,16,19} {5,17} {2,7,10,11,12,13,15,19} {4,6,8,9,16,17,18} {3,7,11,13,14,19} {6,8,10,15,17} {4,5,7,9,12,13,14,18,19} {6,8,15,16} {3,7,10,11,13,17,18} {2,6,12,14,15,19} {4,5,7,11,13,16,17,18} {3,9,12,14,19} {7,8,10,11,13,15,16} {3,5,6,9,12,14,17,18,19} {2,13,15,16} {5,6,9,10,14,18,19} {1,12,13,15,17} {2,6,7,11,14,16,18,19} {10,12,13} {2,4,8,14,17,18,19} {9,11,12,15} {3,4,8,10,13,14,17,18,19} {5,11,12,15,16} {6,7,8,13,18,19} {5,10,11,12,14,15,16} {4,7,8,9,13,17,18,19} {6,11,15,16} {1,7,8,13,14,17,18,19} {9,10,11,12,15,16} {7,8,13,14,19} {5,6,10,11,12,15,16,17,18} {7,9,14,19} {2,11,12,13,15,16,17} {9,10,18} {3,7,11,12,14,16,17,19} {6,10,13,15,18} {3,9,11,12,16,17,19} {8,13,14,15} {4,7,11,12,17,18,19} {5,13,14,15,16} {6,8,10,11,12,17,18,19} {9,14,15} {7,10,11,12,13,17,19} {3,14,16,18} {7,11,12,15,19} {5,10,13,14,17,18} {8,9,11,12,15,16,19} {7,14,18} {6,8,13,16,17,19} {9,10,14,15,18} {6,11,17,19} {5,12,13,14,15,16,18} {10,11,19} {7,13,14,15,18} {9,10,11,12,16,17,19} {13,14} {8,12,15,16,17} {9,13,19} {10,11,12,14,15,16,18} {6,13,17,19} {8,12,15,16,18} {9,11,13,14,17,19} {10,18} {8,12,13,16,17,19} {10,14,15,18} {11,13,19} {10,14,15,16,17} {8,13,18,19} {11,14,15,16,17} {7,18,19} {11,15,16,17} {10,13,14,18,19} {12,16,17} {11,13,18,19} {14,15,17} {11,13,16,18,19} {14,17} {12,16,19} {13,15,17,18} {16} {12,18,19} {16,17} {13,18,19} {17} {16,19} {18} {19}
L(20) = 277:  peak size 12 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,9} {2,3,4,6,11} {1,5,7,8} {3,4,9} {1,2,5,6,13} {4,7} {2,3,6,8,9} {1,4,12} {2,5,6,10} {1,3,4,8,13} {2,7,9} {3,4,5,6,8,11} {1,2,7,9,12} {3,5,6,11} {1,2,4,7,8,10,12} {3,5,11} {1,4,8,9,10} {2,3,5,16} {1,4,6,7,8,9,11} {2,3,5,10,13} {1,7,12} {2,3,4,6,8,11,13} {1,5,9,12} {2,3,6,10,13} {1,5,7,8,9,11} {3,6,12} {1,4,5,7,8,10,15} {2,3,6,9,11,12} {1,5,7,10,13} {2,4,6,17} {1,3,7,8,9,10,13} {2,6,14} {1,3,4,7,8,11,12,13} {2,5,6,10,14} {1,3,4,7,8,11,18} {2,5,9,14} {1,3,6,8,10,12,13} {2,4,7,11,14} {1,3,5,6,8,10,13,15} {2,9,12} {3,4,5,7,8,13,14} {1,2,11,17} {3,4,5,6,8,9,10,12,13} {1,2,7,11,18} {3,4,5,8,12,15} {1,2,6,7,10,13,16} {3,4,11,14} {1,2,5,7,9,10,15,16} {3,4,6,8,11,12,13} {1,2,5,7,15,19} {3,4,6,8,20} {1,2,5,7,9,10,11,13,16} {3,4,6,20} {1,2,5,9,12,13,16} {6,8,11} {1,2,3,7,9,13,14,18} {5,6,8,11,12} {2,3,4,9,10,14,17} {6,7,8,13} {1,2,3,5,9,11,14,15,17} {4,6,7,8,10,12,13} {1,2,3,17,20} {4,5,6,7,8,10,11,18} {1,9,16} {2,3,4,8,13,14,17} {1,7,9,18} {2,3,6,8,10,11,12,13,14} {4,5,7,9,19} {2,3,6,11,13,18} {1,4,5,7,9,17,19} {2,3,6,8,11,12,13,16} {4,5,7,10,19} {2,3,6,8,9,11,12,15,16} {1,4,5,7,10,13,14,19} {2,3,6,8,12,16,17} {1,4,7,9,14,20} {2,3,5,6,8,10,11,12,13,15,16} {1,4,7,14,20} {2,3,5,6,11,12,13,15,16} {1,8,10,18} {2,3,4,7,13,17,19} {1,5,6,8,9,10,11,12,15,16} {2,3,4,13,14,20} {1,5,7,9,10,11,15,17} {3,4,8,13,19} {1,2,5,11,12,15,20} {3,4,7,8,9,10,13,14,17} {1,2,15,20} {4,5,7,8,9,10,12,14,17} {1,2,3,6,16,19,20} {4,7,9,10,18} {1,3,6,8,13,14,15,17} {4,5,7,11,12,19} {1,2,3,6,9,10,14,15,17,20} {4,5,7,11,12,13,16} {1,2,3,6,8,9,10,14,17,18,19} {5,7,11,16} {2,3,6,8,9,10,13,17,20} {1,4,7,18,19} {2,3,5,8,9,10,11,13,14,16,17} {1,4,7,12,15,20} {3,5,9,10,11,13,18} {1,2,6,8,12,14,17,19} {3,4,5,9,10,11,13,16,18} {2,6,8,15,19} {1,3,4,5,9,12,14,16,17,20} {2,6,7,8,10,11,13,15,19} {1,3,5,9,12,16,17,18} {4,6,8,11,13,14,15} {1,5,9,10,16,20} {2,3,4,7,8,11,12,13,15,18,19} {1,9,10,14,17} {2,3,4,6,11,13,16,18,19} {1,8,15,17} {2,3,4,6,7,9,11,14,18,19,20} {1,5,8,12,13,16,17} {2,3,4,7,9,10,14,15,19,20} {1,13,17} {2,3,4,8,9,10,12,14,15,18,19} {1,5,6,7,11,16,17,20} {4,8,9,12,19} {2,3,5,6,7,10,11,13,14,16,18,20} {1,4,8,9,15,17,19} {5,6,7,10,11,12,13,14,16} {2,8,15,17} {4,5,6,7,9,10,11,12,14,18,20} {2,3,8,13,15,16,17} {1,4,5,6,7,10,11,12,14,18,19,20} {2,3,15,16,17} {4,6,7,11,12,13,14,18} {1,2,3,5,9,10,15,16,17,19,20} {8,11,13} {1,3,5,9,10,12,15,16,17,19} {4,7,8,11,13,14,18} {1,3,5,6,9,10,12,16,17,19,20} {7,8,13,15} {1,2,9,10,11,12,16,17,19} {4,7,8,15,20} {2,3,5,9,10,12,13,14,16,17,18} {4,7,8,11,15,20} {1,2,6,14,16,18,19} {3,4,9,11,12,13,15,20} {1,6,7,8,10,14,16,17,19} {2,4,5,9,11,12,13,15,18,20} {6,8,19} {2,3,4,7,11,13,16,17,18,20} {5,6,8,10,12,14,15,19} {1,7,9,11,13,17,20} {2,6,12,14,15,18} {1,4,5,7,10,11,13,16,17,19,20} {6,9,12,14,15} {2,3,4,8,11,16,18,19,20} {6,7,15,17} {4,5,8,9,10,11,12,13,14,18,20} {2,3,7,15,16,17,19} {4,5,8,9,10,12,13,14,18,20} {3,15,16} {1,5,9,10,11,17,18,20} {2,7,14,15,19} {1,8,10,11,12,18,20} {3,4,7,9,13,15,16,17,19} {1,8,10,12,18,20} {2,3,7,11,14,15,16,17,19} {4,6,8,12,13,18,20} {1,10,14,16,17} {6,7,8,9,12,13,18,20} {3,15,17} {1,4,8,9,11,13,14,18,19,20} {5,7,10,12,15,16,17} {4,9,14,20} {1,6,7,11,13,15,16,18,19} {4,8,10,17,20} {5,6,7,9,11,12,13,14,16,18,19} {1,8,10,15,17,20} {2,9,11,12,14,16,19} {5,6,8,10,13,15,18,20} {2,7,9,11,12,14,16,17,19} {1,8,13,18,20} {2,6,9,11,14,15,16,17,19} {5,7,10,12,13,18,20} {4,8,11,15,16,19} {7,10,12,14,18} {4,5,6,8,15,16,17,19,20} {7,11,13,18} {3,6,9,14,15,19,20} {5,8,10,16,17,18} {2,4,12,13,14,15,19,20} {7,10,11,16,18} {4,6,12,14,15,17,19} {7,8,9,10,11,13,16,18,20} {6,12,15,17} {2,7,11,13,16,19,20} {4,12,14,15,18} {5,7,9,11,13,17,19,20} {3,10,14,15,16,18} {6,7,9,11,17,19,20} {3,14,16,18} {4,9,10,11,12,13,17,19,20} {1,14,15,16,18} {2,11,12,13,19,20} {6,8,10,15,16,17,18} {5,9,11,12,13,14,19,20} {7,15,17} {2,10,12,13,16,19,20} {7,8,14,15,17,18} {3,11,13,19,20} {8,9,10,12,14,15,16,17,18} {1,13,19,20} {4,11,12,15,16,17,18} {7,14,19} {5,9,10,13,15,17,18,20} {7,11,14,16,19} {5,6,13,15,17,18,20} {8,19} {6,11,13,14,16,17,18} {4,10,15,19,20} {8,11,12,13,14,16,17,18} {2,19,20} {9,10,12,14,16,17,18} {3,13,19,20} {8,10,16,17,18} {6,11,13,14,19,20} {9,10,12,15,16,17,18} {6,11,19,20} {10,13,14,15,16,17} {5,9,18,19,20} {10,14,16,17} {8,9,11,15,18,19,20} {10,16,17} {8,11,14,19,20} {10,12,13,15,16,17,18} {5,14,19,20} {12,13,15,16,17} {5,19,20} {10,13,14,15,16,17,18} {9,11,19,20} {12,13,14,17,18} {8,11,15,16,19,20} {12,18} {10,11,14,17,19,20} {12,15,16,18} {9,17,20} {12,13,15,18,19} {9,16,17,20} {12,14,15,18,19} {10,17,20} {14,15,16,18} {11,12,17,19,20} {14,18} {12,13,17,19,20} {15,16,18} {12,14,17,19,20} {15,18} {14,17,19} {13,16,18,20} {17} {14,16,19,20} {17,18} {14,19,20} {18} {17,20} {19} {20}
L(21) = 321:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,4,5,9} {1,6,7} {2,3,4,10} {1,9} {2,4,5,7,8} {1,3,6,11} {4,5,7} {1,2,3,6,15} {4,7} {1,2,3,8,14} {4,6,7} {2,3,5,13} {1,4,6,7,11} {2,10} {1,4,5,6,8,13} {2,3,7,9,10} {1,4,8,12} {2,3,5,6,7,10,11} {1,4,14} {2,6,7,8,9} {1,4,5,16} {2,3,6,7,8,9,11} {1,4,5,10,13} {2,3,15} {1,4,6,7,8,9,12} {2,3,5,17} {4,6,7,8,9} {1,2,3,5,13,17} {7} {2,3,4,8,9,11,13} {1,5,6,7,10,14} {2,3,4,8,19} {1,5,6,17} {2,3,4,7,10,11,14} {1,9,12} {2,3,5,6,7,8,11,17} {1,4,9,10,13} {2,3,5,6,7,14,15} {1,4,12,13} {3,5,7,8,9,10,11} {1,2,4,14,17} {3,5,6,7,8,9,11,12} {1,2,4,10,14,15} {3,5,6,8,9,11,12} {1,4,7,19} {2,3,5,6,8,9,11,12,14} {1,4,7,10,17} {2,5,6,9,11,14} {1,3,4,8,10,13,16} {5,6,7,14} {1,2,3,8,9,12,13,17} {4,5,6,7,10,11,14} {2,3,8,9,12,15} {1,6,10,11,13} {2,3,4,5,7,8,9,15,21} {1,6,12,14} {3,4,5,7,8,13,18} {1,9,15} {2,3,7,8,10,11,12,14} {1,4,9,13,15} {2,3,6,7,11,14,16} {1,8,12,13} {2,3,4,7,9,10,11,14,17} {1,5,6,8,12,13,15} {2,4,9,11,17} {1,5,6,7,8,10,14,18} {3,11,12} {1,2,4,5,6,8,10,14,16,21} {3,7,12,13} {1,4,6,8,9,10,11,14,16} {2,3,7,13,19} {1,5,9,10,11,17} {2,4,6,7,12,15,16} {1,5,8,9,10,11,13,14} {2,4,7,12,20} {1,3,5,6,10,11,13,14,19} {2,4,7,8,9,12,15,16} {1,3,5,10,13,14,18} {2,4,7,9,16,17} {1,3,5,6,8,10,11,12,13,14,18} {2,4,9,15,16} {1,3,5,6,7,11,12,18,20} {4,8,9,16} {1,3,5,7,11,17,21} {2,4,6,8,9,10,12,13,14,15} {3,5,20} {1,4,6,7,9,12,17,19} {2,5,11,14,15} {4,6,8,9,10,12,17} {1,2,3,5,11,14,15,16,18} {4,6,8,10,12,17} {1,3,5,7,9,11,13,16,21} {2,4,6,8,12,15,20} {1,7,10,11,19} {2,5,6,8,9,14,16,17} {1,7,21} {2,3,4,5,6,9,14,15,19,20} {1,7,8,10,12,13,17} {2,3,4,5,6,9,11,14,15,18,20} {1,8,13,17} {4,5,6,7,9,10,11,16,20} {2,3,12,15,17} {1,4,5,7,9,10,11,13,14,16,18} {2,3,8,12,15,19} {1,4,6,11,13,16,18} {2,5,7,9,10,12,15,19} {1,3,4,6,8,11,13,14,18,21} {5,7,9,19} {1,3,6,8,10,11,14,15,16,17} {2,4,5,7,9,12,13,18,21} {1,3,8,10,11,15,16,17} {5,6,7,9,12,14,18} {1,4,8,11,17,20} {2,3,5,6,7,9,12,15,16,18,19} {1,4,11,14,21} {3,5,6,7,8,10,16,18,19} {1,2,17,21} {4,5,7,8,9,10,11,12,13,16,18} {1,2,3,6,19,20,21} {4,5,7,8,9,10,13,14,16,17} {1,2,6,15,18,20} {3,4,5,7,8,10,11,12,16,17,21} {1,2,6,9,13,15,18,19} {5,7,10,14,16} {1,2,3,4,8,9,12,13,15,17,20,21} {5,6,7,10,11,16,18} {2,4,8,9,12,13,14,15,17} {10,11} {2,3,4,6,7,8,14,15,17,19,21} {1,5,9,10,11,12,13,16,18} {2,4,7,8,15,17,21} {1,3,5,6,9,10,11,13,14,16,19,20} {2,7,8,15,21} {3,5,9,10,11,12,16,19} {1,2,4,6,7,13,14,15,17,18,20} {5,8,19} {2,3,7,10,11,12,13,14,17,18} {1,4,5,9,16,19,21} {2,3,6,7,10,11,13,14,15,17,20} {5,8,9,21} {1,2,6,10,12,13,16,17,20} {3,4,14,15,18} {1,2,5,6,8,11,13,16,17,19,21} {3,4,9,14,15,20} {1,2,5,6,8,10,11,12,13,16,17,19,21} {3,4,7,9,15,18,20} {1,5,8,10,12,13,17,21} {2,4,7,9,11,14,15,16,20} {1,5,6,8,10,12,13,17,18,19} {4,9,20} {1,6,7,8,12,13,14,15,17,18} {2,3,5,9,10,11,19,20,21} {1,6,8,12,13,15,16,18} {2,3,7,11,14,20,21} {1,4,5,6,9,10,12,13,15,16,17,18,19} {2,3,7,14,20,21} {1,5,6,8,10,12,13,15,16,18,19} {3,4,11,17,21} {1,2,7,9,12,13,18,19,20} {8,10,11,16} {1,3,4,6,7,12,14,17,19,20,21} {5,8,9,10,13,16,18} {2,4,6,7,11,12,14,17,19,21} {8,10,16} {1,3,4,5,9,13,15,17,18,19,21} {2,6,10,11,12,14,16,20} {4,8,13,15,17} {1,2,6,7,9,11,12,14,16,18,20,21} {3,5,8,13,15,17,19} {1,4,6,10,11,14,16,20,21} {3,5,7,8,9,12,13,15,17,18,19} {6,10,14,16} {2,4,8,9,11,13,17,19,21} {1,5,10,14,15,16,20} {2,3,6,8,9,11,12,13,17,18,19,21} {1,7,14,16,20} {3,4,6,10,13,17,19,21} {2,5,7,8,11,12,14,15,16,18,20} {6,10,19} {1,3,8,12,13,14,15,16,17,18} {2,4,5,11,19,20,21} {1,6,8,9,12,13,14,15,16,17,18} {2,4,20,21} {1,6,9,11,12,14,16,18,19} {3,5,13,17,21} {4,6,7,8,11,12,14,15,16,18,19} {1,2,10,17,20,21} {7,8,9,11,14,15,19} {2,5,6,12,13,16,20,21} {3,8,9,10,11,14,15,18,19} {2,5,6,7,12,13,16,17,20,21} {9,10,11,18} {2,3,6,8,12,14,16,19,20,21} {4,9,10,11,13,15,17,18} {1,3,5,16,19,20,21} {8,9,11,12,15,18} {1,4,6,7,10,13,16,17,19,20,21} {3,11,14,15,18} {4,7,8,9,10,16,17,19,20} {1,13,14,21} {2,5,8,10,11,12,15,16,17,19,20} {4,7,9,13,14,18,21} {2,5,6,11,12,15,16,17,19,20} {4,8,9,14,18,21} {3,5,7,10,11,13,15,16,17,19,20} {2,9,12,18,21} {4,7,8,13,16,19,20} {1,3,11,12,14,15,17,18,21} {5,10,16,19} {4,6,7,11,14,15,17,18,21} {1,9,10,13,16,19,20} {4,7,14,17,21} {2,8,10,12,15,16,18,20} {1,4,14,17,19,21} {7,8,9,10,11,13,15,16,18,20} {2,4,12,14,17,19,21} {3,13,15,20} {5,8,9,11,12,16,17,18,19} {4,6,13,20,21} {7,8,9,10,11,12,14,16,17,18,19} {3,5,13,15,20,21} {1,9,12,14,17,18,19} {5,7,8,11,15,16,20,21} {4,9,10,12,13,14,17,18,19} {8,15,16} {5,7,11,12,14,17,19,20} {1,8,13,15,16,18,21} {3,11,12,14,19,20} {6,7,15,17,21} {4,9,10,11,12,16,18,19,20} {7,14,15,17} {1,4,13,16,18,20,21} {9,12,19} {4,7,10,13,14,18,20,21} {8,9,15,16,19} {7,10,12,13,14,17,21} {5,8,9,11,15,16,18,19,20} {4,12,17,21} {3,6,13,16,18,19,20} {7,11,14,15,21} {4,9,10,12,17,18,19,20} {8,11,13,14,15,21} {1,9,12,17,18,19,20} {10,11,13,21} {4,9,12,14,15,16,17,18,19} {7,10,11,20,21} {4,12,15,16,17,19} {5,9,11,13,18,20,21} {8,10,12,14,15,16,17,19} {2,9,18,20,21} {10,12,13,14,15,16,19} {4,5,17,18,20,21} {9,13,14,16,19} {7,8,10,12,15,17,18,20,21} {11,13,14,19} {6,7,15,16,17,18,21} {9,14,20} {8,11,12,13,15,16,19,21} {3,14,17,18,20} {8,10,12,15,16,19,21} {3,17,18,20} {7,11,13,15,16,19,21} {10,12,14,17,20} {8,9,11,15,16,18,19,21} {10,14,20} {8,9,13,16,17,19,21} {10,14,15,20} {1,16,17,19,21} {11,12,13,15,18,20} {8,9,14,16,17,19,21} {12,18} {7,9,15,19,20,21} {12,13,16,17,18} {7,15,19,20} {10,12,13,16,17,18,21} {11,15,20} {12,13,16,17,19} {9,11,14,15,18,20,21} {10,16,17,19} {8,14,15,20,21} {12,17,18} {11,13,14,15,16,20,21} {9,17,18,19} {10,13,15,20,21} {11,14,16,17,18,19} {10,13,20,21} {11,16,17,18,19} {9,15,20,21} {12,18,19} {10,14,17,20,21} {15,18} {13,14,16,19,21} {12,18,20} {11,16,19,21} {14,15,17,18,20} {13,21} {14,16,19,20} {13,18,21} {15,16,17,19,20} {14,21} {16,18,19} {13,17,20,21} {18} {15,17,20,21} {18,19} {15,20,21} {19} {18,21} {20} {21}
L(22) = 369:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,5,7} {4} {1,3,5,8} {2,4,7} {1,8} {2,3,5,6,7} {1,4,9} {2,3,6,8} {1,9} {2,4,5,7,8} {1,3,6,11} {2,5,9} {1,3,4,8,11} {5,6} {1,2,4,7,14} {3,6,8} {1,4,5,13} {2,3,6,8,10} {1,4,7,12} {2,3,5,6,8,13} {1,4,7,9,10} {3,5,6,11} {1,2,4,7,8,9,13} {3,5,11} {1,4,6,8,13} {2,5,7,12} {1,3,4,8,9,10,11} {2,5,6,7,13} {1,3,16} {2,4,5,6,7,9,14} {1,3,10,13} {2,4,6,8,14} {1,3,5,7,12,13} {2,4,6,9,14} {1,3,5,7,10,11,13} {2,4,6,8,9,14} {1,3,5,11,16} {2,4,6,17} {1,3,5,9,10,11,12} {2,6,14} {1,3,4,5,8,10,12,16} {2,6,7,9,13} {1,3,4,5,8,11,20} {6,9} {1,2,3,8,11,13,15} {4,5,7,10,12} {1,2,3,6,8,11,14,16} {4,9,10} {2,3,5,7,8,14,15} {4,6,9,12} {1,2,3,5,8,10,11,14,16} {4,6,7,9,13} {1,2,3,10,14,17} {4,5,6,7,8,12,13} {1,3,20} {2,4,6,8,9,11,12,13} {1,3,5,7,10,14,17} {2,4,6,9,13,15} {1,3,7,11,19} {2,4,5,6,8,10,12,13,14} {1,3,7,22} {2,4,5,9,10,13,15} {3,8,14} {1,4,5,6,11,12,13,15} {2,3,8,10,19} {1,5,6,7,9,15,16} {2,8,11,13} {1,4,5,7,9,10,12,14,15} {2,3,6,8,11,13,17} {4,5,7,12,15} {1,2,3,8,10,11,13,21} {4,5,17} {1,2,3,6,8,9,10,13,15,20} {5,7,11,12} {1,2,3,4,9,10,14,16,20} {5,6,8,12,13} {1,2,3,7,19,21} {4,5,6,8,9,12,18} {1,2,3,7,10,11,13,14,19} {4,5,18} {1,2,7,8,10,12,13,14,15} {3,4,5,6,9,11,17,18} {1,2,7,8,10,15,21} {3,4,9,12,13,14} {1,2,5,6,7,8,10,11,16,17,18} {3,4,12,13,14} {1,2,5,6,10,11,15,16,17} {4,7,8,18} {2,5,6,9,11,12,20} {1,3,4,7,8,10,13,14,15,18} {2,6,9,11,12,16} {1,3,5,8,10,13,14,21} {2,4,6,15,20} {1,7,8,9,11,13,17} {2,3,4,5,6,10,15,19,21} {8,11} {2,3,4,6,7,10,16,17,21} {1,5,8,9,11,13,20} {2,3,7,14,22} {1,4,5,6,9,15,16,21} {7,10,12} {1,3,5,6,8,9,11,15,17,22} {2,4,7,10,13,14,18} {1,3,5,6,8,9,11,12,16,17,19} {2,4,13,20} {1,3,6,7,8,9,14,18,22} {2,5,10,15,17} {1,3,4,6,7,9,11,14,16,18,19} {2,5,8,10,12,22} {1,3,6,11,14,16,18} {4,5,7,8,10,12,13,20} {1,2,3,6,9,14,16,17,21} {4,5,13,18} {1,2,3,7,10,12,14,16,17,19} {4,5,6,8,9,11,13,15,20} {1,2,3,7,14,16,17,21} {5,6,8,9,10,11,22} {1,3,4,15,17,21} {2,5,6,7,8,11,12,13,14,16,18} {1,3,9,17,21} {2,4,6,8,10,13,15,16,18} {3,7,9,22} {1,4,5,6,8,11,12,14,16,17,19} {2,3,7,9,10,20,21} {1,4,5,6,8,12,14,15,16,22} {2,10,19} {1,4,5,6,8,12,13,14,16,17,18} {2,3,7,9,10,11,20,21} {5,8,12,13,14} {1,2,3,4,6,7,9,15,18,19,20,21} {5,8,10,11,12,13,14} {1,2,4,6,9,15,16,19,22} {3,5,8,10,17,20} {1,2,4,6,7,11,12,15,18,19,21} {3,5,8,9,10,13,14,16,17} {2,4,6,7,12,21,22} {1,3,5,8,9,10,11,13,14,16,18,19} {2,4,12,15,20} {1,3,8,9,11,13,18,22} {2,4,5,6,7,12,14,15,16,17,19} {9,10,13} {1,3,5,6,11,12,15,16,18,20} {2,4,8,9,13,17,22} {1,3,5,6,7,11,14,15,16,19,21} {2,10,13,18} {1,4,5,7,11,12,17,19,21} {2,3,13,14,22} {1,5,6,7,9,10,11,12,18,19,21} {2,3,8,15,17,20} {1,4,5,6,7,9,11,12,13,14,18,19,22} {2,3,8,10,16,17,20} {1,4,5,12,13,15,18,19} {2,3,6,8,10,11,16,20,22} {1,4,5,9,12,13,14,15,17,19} {3,8,10,16,18} {1,2,5,6,9,12,13,14,17,21,22} {3,4,7,8,10,11,15,16,18,19} {2,5,6,9,12,13,14,17,22} {1,3,4,8,15,18,19,21} {5,6,7,11,13,14,22} {1,2,3,4,8,9,10,15,16,17,19,20,21} {6,7,11,12,13,18} {1,2,5,8,9,10,14,15,16,21,22} {3,4,13,17,19} {2,5,6,9,11,14,15,18,21} {3,4,7,10,13,16,17,20} {1,2,5,8,9,11,14,15,18,19,22} {4,7,10,12,13,16,17} {1,3,5,6,8,11,18,19,20,22} {2,4,10,14,17,21} {1,3,5,7,11,12,13,15,16,20,22} {4,6,8,9,10,17,18,19} {1,2,3,5,7,11,12,13,15,16,20,21,22} {4,8,10,17,18} {2,3,5,6,9,11,12,14,15,19,20,21} {1,7,8,13,16,17,18} {3,4,5,6,11,14,19,20,21} {1,2,8,9,10,12,13,15,16,18,22} {5,6,7,11,19,21} {2,3,4,9,10,12,14,15,16,20,22} {5,6,7,8,11,13,17,18,19} {3,4,9,10,14,20,21} {1,2,5,6,11,12,13,15,16,17,19,22} {7,9,10,14,18} {1,2,3,4,8,11,12,13,15,19,20,21,22} {5,6,7,10,14,16,17,18} {1,2,3,4,11,12,13,19,20,21,22} {6,7,9,14,16,18} {1,3,5,8,12,13,15,19,20,21} {4,6,7,9,16,18,22} {1,2,3,11,12,13,14,15,17,20,21} {5,6,7,8,9,18,19,22} {2,3,10,12,13,14,15,16,21} {4,9,11,17,18} {1,3,5,7,8,14,15,16,19,20,22} {2,9,12,13,17,18} {1,4,6,7,8,10,11,14,15,16,19,21,22} {2,5,9,12,17,18,20} {4,6,8,10,11,13,21,22} {1,2,3,14,15,16,17,19,20} {6,7,8,9,10,11,12,13,21,22} {1,2,4,5,14,15,16,17,18,19,20} {6,9,11,22} {1,4,5,8,10,16,17,19,20,21} {6,7,9,11,12,13,14,15,22} {2,3,8,10,17,18,19,20} {5,6,9,12,15,16,22} {1,2,4,8,10,11,13,14,17,18,19,20,21} {5,6,9,15,16,22} {2,3,4,11,12,13,14,17,18,19,21} {7,8,10,16,20} {1,4,9,11,13,15,17,19,21} {2,5,20,22} {1,3,7,10,12,13,15,16,18,19,21} {2,5,6,14,17,20,22} {1,4,9,10,12,13,16,18,19,21} {6,11,20} {2,3,7,9,10,13,15,16,18,21,22} {4,5,8,12,14,17,19,20} {1,2,16,21,22} {4,6,8,9,11,12,13,14,15,18,19,20} {1,3,7,16,17,21,22} {4,5,9,10,13,14,18,19,20} {2,6,7,8,11,12,15,16,17,21,22} {4,13,14,19} {1,2,5,7,11,15,16,18,20,21,22} {6,9,10,13,14,17,19} {3,4,5,7,8,12,15,16,18,20,21,22} {2,11,14,17,19} {1,5,9,10,13,20,21,22} {2,7,8,11,12,14,15,16,17,18,19} {1,3,9,20,21,22} {6,7,8,10,13,14,15,17,18,19} {4,5,11,12,16,20,21} {1,6,7,8,13,14,15,17,18,19,22} {4,11,16,20} {1,5,7,12,14,15,18,21,22} {2,9,16,17,20} {4,5,8,10,11,12,14,15,19,21,22} {1,9,13,16,18,20} {6,7,10,12,15,19,21} {1,2,13,14,16,17,18,22} {5,6,8,10,12,15,19,20,21} {2,7,9,11,13,14,16,17,18,22} {3,10,12,19,21} {4,8,9,11,15,16,17,18,20} {1,3,12,13,14,19,21,22} {7,9,10,11,17,18,20} {4,12,13,14,15,21} {2,3,7,10,11,16,17,18,19,20,22} {8,9,13,15,21} {2,6,10,11,12,17,19,20,22} {3,14,15,21} {4,5,6,9,10,16,17,18,19,20,22} {7,11,12,13,14,15,21} {1,2,8,10,16,17,18,19,20,22} {9,11,12,13,14,21} {3,4,8,15,17,18,20,22} {6,12,14,16,19} {3,8,11,13,17,20,22} {5,6,10,12,14,16,18,19,21} {2,15,17,20} {5,8,9,12,14,16,18,19,21} {4,10,11,13,15,20,22} {5,6,17,19,21} {8,11,12,13,14,15,16,20} {2,17,22} {4,7,11,13,14,15,16,18,19,20} {6,8,10,12,17,21,22} {7,14,16,18} {3,8,9,12,15,17,19,20,21} {6,7,16,18,22} {4,9,10,11,12,14,15,17,19,20,21} {6,8,13,16,18,22} {2,4,14,17,19,20,21} {5,10,12,13,15,16,18,22} {6,8,9,11,14,17,19,20,21} {7,15,16,18} {6,9,10,12,13,20,21,22} {8,11,14,15,16,17,18} {4,9,12,19,20,21} {7,13,16,17,18} {2,6,9,14,15,19,20,21,22} {11,13,16,17} {6,8,12,15,18,20,21} {3,11,14,17,19,22} {7,8,10,15,16,18,20,21} {9,13,14,17,19} {6,8,10,16,18,21,22} {9,11,12,13,14,15,17,19,20} {2,16,18,22} {4,6,15,17,19,20,21} {10,11,12,18,22} {6,7,13,14,17,19,20,21} {8,9,15,16,18,22} {5,11,13,14,19,20,21} {10,12,15,22} {5,9,13,14,16,17,18,20,21} {7,11,15,19,22} {9,12,14,16,17,21} {7,8,10,18,19,20,22} {11,12,13,14,15,16,17,21} {2,9,18,19,20,22} {11,12,14,15,16,17,21} {5,7,18,19,20,22} {12,13,14,16,21} {7,9,10,15,17,18,19,20,22} {11,13,16,21} {4,10,14,18,19,20,22} {11,12,15,16,17,21} {7,10,13,14,18,19,20,22} {8,15,16,17,21} {2,13,14,18,19,20,22} {10,21} {5,13,14,17,18,20,22} {11,12,15,19,21} {8,10,14,16,17,18,20,22} {13,15,19} {3,14,16,17,18,20,22} {11,12,19,21} {9,10,18,20,22} {11,13,14,17,19,21} {8,12,15,16,18,20,22} {9,14,17,19,21} {8,13,16,18,20,22} {9,15,17,19,21} {11,14,18,22} {6,15,16,17,19,20,21} {9,18,22} {13,14,15,19,21} {10,12,16,17,18,20,22} {11,15,19,21} {12,16,17,18,20} {10,19,21} {12,14,15,16,18,20,22} {10,17,19,21} {13,14,15,20,22} {10,16,17,18,19,21} {9,20,22} {13,15,17,18,19,21} {11,16,20,22} {13,18,21} {15,16,17,19,20} {14,21} {15,16,18,19,20} {10,21,22} {15,17,19,20} {12,16,18,21,22} {17,19} {14,16,21,22} {17,18,20} {14,16,19,21,22} {17,20} {15,19,22} {16,18,20,21} {19} {16,18,21,22} {19,20} {16,21,22} {20} {19,22} {21} {22}
L(23) = 419:  peak size 13 -- {1} {2} {1,4} {3} {1,2,7} {3,4} {1,2,8} {4} {2,3,5,7} {1,4,8} {2,7} {1,3,4,6,9} {2,5,7} {1,3,4,6,10} {2,8} {1,4,5,7,9} {2,3,6,10} {4,5,7} {1,2,3,8,13} {4,7} {1,2,5,6,14} {3,4,10} {1,5,6,11} {2,3,4,7,9,10} {1,11} {3,4,6,7,8,9} {1,2,5,10,13} {3,4,6,12} {1,2,5,7,8,10,11} {4,6,9} {1,2,5,10,14} {6,7} {1,3,4,8,9,10,11} {2,5,6,7,13} {1,8,11} {2,3,5,6,7,9,15} {1,4,8,14} {3,5,6,9,11} {1,2,4,8,10,16} {3,11} {1,4,6,8,9,10,12} {2,3,5,7,11,15} {1,4,8,10,13} {3,5,9,12} {1,2,4,8,10,11,15} {3,6,13} {1,2,4,7,8,9,11,17} {3,5,6,10,13} {1,2,4,8,9,12,16} {3,6,10,11} {1,2,4,7,9,13,17} {3,5,8,10,12} {1,2,4,6,7,9,15,17} {5,8,10} {1,3,4,6,7,15,18} {2,5,11,13} {1,3,4,7,8,9,10,12,16} {2,5,6,11,15} {1,3,4,10,12,17} {2,5,6,7,9,11,15} {1,3,4,14,18} {2,6,7,8,9,10,11,12} {1,3,4,5,13,14,17} {2,7,8,9,11,12} {1,3,6,13,18} {2,4,5,7,8,9,10,14,15} {1,3,12,17} {2,5,8,9,10,11,13} {1,4,20} {2,3,5,6,8,12,15,16} {1,7,9,11,14} {2,3,4,5,10,12,23} {6,11} {1,3,4,5,8,10,14,15,17} {2,6,7,9,11,12,13} {1,5,8,14,15} {2,3,4,6,9,11,12,22} {1,5,7,8,10,21} {2,3,4,9,13,14,16} {1,5,10,19} {3,4,6,7,8,9,12,13,17} {1,2,5,16,20} {3,6,8,11,12,13} {2,4,7,9,10,14,16} {1,3,5,6,8,12,13,23} {7,11} {3,4,5,6,8,9,12,16,19} {1,2,7,10,11,13,14,15} {3,4,5,6,9,16,21} {1,7,8,10,12,17} {2,3,4,5,6,9,11,13,14,15,19} {1,7,8,12,18} {2,3,4,5,6,9,14,17,23} {1,7,11,18} {2,5,9,10,12,13,14} {1,3,4,6,7,8,11,16,17,20} {2,12,14} {1,4,6,7,11,13,15,18} {2,3,5,17,20} {1,6,7,9,12,15,16} {2,3,4,5,8,10,11,19,23} {1,18} {3,5,6,7,8,11,12,14,20} {1,2,4,9,10,18,23} {5,7,8,13,15} {2,3,4,6,12,14,17,19} {1,5,7,8,15,22} {2,3,4,6,9,10,11,16,17,19} {1,5,7,8,12,14,21} {2,3,4,6,9,10,11,13,15,16,18} {1,7,12,19} {2,3,4,6,10,11,14,17,21} {1,5,9,12,22} {2,3,4,6,8,10,11,13,14,17,20} {1,5,7,12,16,18} {3,4,8,10,11,13,20} {1,2,5,6,9,16,18,22} {3,4,7,8,10,11,12,14,20} {2,6,22} {4,5,7,8,9,10,11,14,16,17} {1,2,3,6,12,13,15,18,21} {5,7,8,9,10,11,14,17} {1,2,3,12,13,19,21} {4,6,7,10,16,18} {1,2,3,5,8,9,11,13,17,20,23} {4,6,12,14,15} {2,3,7,8,10,11,13,17,21} {5,9,12,15} {1,2,3,4,7,8,13,17,18,19,21} {5,6,9,10,11,15,16} {1,2,3,4,7,12,14,18,20,22} {5,6,9,10,11,21} {1,2,3,4,7,8,12,17,18,19,23} {5,6,9,10,11,13,14,15} {1,4,7,17,23} {2,3,5,6,8,9,10,12,13,18,19,20} {1,4,7,11,14,15,21} {2,3,5,6,8,13,16,19,22} {1,10,14,17} {2,5,7,8,9,11,12,13,15,16,18} {1,3,4,6,10,14,17,19,21} {2,5,8,11,12,16,20} {1,3,4,6,7,9,10,13,15,17,19,23} {2,5,8,16,22} {1,3,7,10,12,14,15,23} {2,4,5,6,8,9,13,16,17,18,19} {7,11,14} {1,2,3,5,6,15,16,17,20,22} {4,7,9,10,12,14,19} {1,2,3,5,6,11,15,16,18,20,21} {4,7,9,23} {2,5,8,10,11,13,15,16,17} {1,3,9,20,21} {2,4,7,8,10,11,12,13,16,17,19} {1,5,6,14,18,21} {2,3,4,7,8,9,10,11,13,15,17,19,23} {1,5,6,12,14,16,22} {3,4,7,9,13,15,17,19} {2,5,6,10,11,12,14,18,20} {1,3,4,7,9,13,15,17,19,21} {2,8,23} {1,3,6,7,10,12,13,14,16,19,21} {2,4,5,8,9,11,15,17,18,22} {1,3,7,10,12,13,14,19,21} {2,4,5,6,15,16,18,23} {3,7,8,9,13,17,21} {1,2,4,5,6,10,11,14,15,16,19,20,22} {3,7,9,12,13,23} {1,2,4,6,11,14,15,16,17,18,19} {5,7,8,13,23} {1,2,3,6,14,15,19,20,21} {4,7,16,18} {1,3,5,6,9,12,13,15,19,20,21} {2,4,7,11,14,18,23} {1,5,6,8,9,13,15,17,19,20} {2,3,11,14,16,22} {1,4,6,7,8,10,13,17,18,20,21} {2,5,9,11,12,14,15,23} {1,3,4,6,7,8,10,13,16,18,19,21,22} {2,9,11,12,23} {1,3,4,7,8,10,13,14,16,19,20,22} {2,5,9,11,15,17,21} {1,4,7,8,12,14,16,18,23} {2,3,5,6,10,11,13,15,19,20,22} {1,7,9,12,17,23} {3,4,5,8,10,11,14,15,16,19,22} {1,6,7,9,12,13,17,18,21} {3,5,8,10,14,19,22} {1,2,4,6,9,11,13,15,17,18,20,23} {3,5,12,16,22} {1,2,6,7,8,9,10,13,15,18,19,20,23} {3,4,5,12,14,16,17,22} {1,2,7,8,9,11,13,15,18,21,23} {3,6,10,14,17,20} {1,2,4,8,13,15,16,18,19,21} {5,6,9,11,14,17,20} {1,4,7,8,10,12,13,16,18,19,21} {2,3,20,22} {4,5,6,7,9,10,11,12,14,15,18,19,23} {1,2,3,8,13,16,20,21,22} {6,10,11,15,17} {2,4,5,7,8,12,14,16,19,21,22} {1,6,11,15,18,20} {3,4,5,7,9,10,12,13,14,16,17,21,23} {1,2,6,15,18,19,22} {3,7,8,9,14,16,17,21} {4,5,6,10,12,13,18,19,20} {1,2,3,8,14,15,16,17,21,22} {4,5,6,9,10,11,12,13,18,20,23} {1,8,15,17,19} {3,4,5,6,12,13,14,16,18,20,22} {2,7,8,9,10,11,15,17,19,23} {1,4,5,6,12,18,20,21,22} {2,8,9,13,14,15,17,19} {3,4,7,10,18,21,22} {1,2,5,6,9,11,14,15,16,17,19,20,23} {4,7,10,12,18,22} {2,5,6,8,9,13,15,16,19,20,21} {4,7,10,18,22} {1,6,9,12,13,15,16,17,21} {2,3,4,5,8,10,11,14,18,19,20,22,23} {1,12,15,21} {2,3,4,8,11,13,14,18,19,20,23} {5,7,9,12,15,17,21} {2,4,8,10,11,13,16,18,19,22} {6,7,9,15,17,20} {1,4,5,10,11,12,13,14,21,22,23} {3,7,8,9,15,18,19,20} {1,2,5,6,10,11,13,14,16,17,21,22,23} {7,8,12,15,20} {2,3,5,6,11,13,14,16,17,18,21,23} {1,7,10,12,15,20,22} {3,5,6,9,13,14,18,21,23} {1,4,7,8,11,12,16,17,19,20,22} {5,9,13,23} {1,3,7,8,10,14,15,17,20,21,22} {5,6,9,11,12,13,16,18,23} {4,7,8,14,15,19,21} {1,3,5,6,9,11,16,17,18,20,22,23} {7,10,12,15,19} {2,3,4,5,8,11,14,16,17,18,21,22,23} {6,7,9,12,13,15,19,20} {2,3,4,8,10,11,17,18,21,22,23} {1,12,13,14,16,20} {5,7,8,9,10,11,15,17,22,23} {3,6,12,13,16,19,20} {2,4,7,8,11,14,15,17,18,21,23} {1,9,19,22} {4,6,7,8,12,17,18,20,23} {5,11,13,16,19} {4,6,7,8,9,12,14,15,21,22,23} {2,10,13,16,17,19} {1,3,5,6,7,12,14,15,18,20,21,22,23} {4,10,11,13,16,17,19} {2,6,8,12,14,18,20,23} {3,4,7,10,13,17,19,21,22} {2,5,9,11,12,14,15,18,20,23} {3,6,7,8,10,13,16,17,19,21,22} {4,9,14,15,23} {1,6,8,10,11,16,17,19,21,22} {2,4,9,13,14,15,18,20,23} {5,7,8,11,16,17,19,22} {6,9,10,14,15,18,20} {1,4,5,7,11,12,13,16,17,19,21,22,23} {3,9,14,15,18,20} {1,6,7,10,11,12,13,19,21,22,23} {4,9,16,17,20} {1,2,7,11,14,18,21,22,23} {3,13,17,20} {2,4,7,11,12,14,16,18,19,21,22} {3,5,10,15,17,20,23} {4,7,8,9,12,13,18,19,21,22} {1,10,11,15,20,23} {3,5,6,7,13,16,17,18,19,21,22} {4,9,10,12,14,15,20,23} {1,7,18,19,22} {3,6,9,10,11,12,13,16,17,20,21,23} {2,4,14,15,18,19,22} {3,7,9,12,13,16,17,21,23} {5,6,8,10,11,14,15,18,19,20,22} {4,13,16,21} {1,9,11,12,14,15,17,20,23} {7,8,10,13,16,19,22} {3,4,5,11,12,14,15,17,18,20,21,23} {6,8,13,19,22} {5,9,10,11,14,17,20,23} {1,6,7,12,13,15,16,18,19,21,22} {8,9,11,14,17,23} {1,2,6,12,16,18,19,20,21,22} {4,10,13,14,15,17,23} {3,5,6,9,12,16,18,19,20,21,22} {1,14,17,23} {5,6,11,12,13,16,18,21,22} {7,8,15,19,20} {2,5,6,11,13,14,17,18,21,22,23} {7,10,12,15,19,20} {8,9,11,13,16,18,22} {3,5,12,14,15,19,20,23} {6,8,9,11,13,17,18,21,22} {3,7,10,12,14,15,16,19,20,23} {6,8,17,18,21} {4,5,9,11,12,16,19,20,22,23} {7,8,13,14,15,17,18,21} {4,6,11,16,19,20,23} {7,10,14,15,18,21} {2,6,8,11,12,16,17,19,20,22,23} {4,13,15,18,21} {1,5,11,14,16,17,19,22,23} {6,13,18,20} {4,7,9,11,12,15,16,17,21,22,23} {6,10,13,14,18,19,20} {1,8,9,11,15,16,17,21,22,23} {4,19,20} {5,11,12,13,14,17,21,22} {4,8,18,19,23} {5,10,11,16,17,20,22} {6,7,9,12,15,18,19,21,23} {13,16} {4,5,8,14,17,18,20,22,23} {10,11,12,13,16,19,21} {1,7,20,22,23} {4,12,14,15,16,17,18,21} {3,19,22} {7,8,10,12,15,16,17,18,21,23} {6,9,13,14,19,20,22} {10,11,17,21} {3,7,12,14,15,18,19,22,23} {9,11,16,17,21} {5,7,10,12,13,14,18,19,20,22,23} {9,11,15,16,17,21} {1,5,14,19,20,22,23} {9,11,12,15,16,17,18,21} {6,7,10,13,14,19,20,22,23} {5,16,18,21} {9,11,13,14,15,17,19,23} {3,6,16,18,20,21,22} {11,12,13,15,17,23} {8,14,16,18,20} {9,10,11,13,15,17,19,21,22} {6,12,20,23} {3,13,16,17,18,19,21} {6,10,11,20,22,23} {7,12,14,15,17,18,19,21} {3,9,20,22,23} {10,12,13,16,17,19,21} {7,9,11,14,15,18,20,22,23} {5,17,19,21} {6,13,14,15,18,20,23} {9,11,17,19,22} {8,10,12,15,16,20,21,23} {11,14,22} {8,10,13,17,18,21,23} {7,15,19,22} {9,12,13,14,16,17,18,20,23} {6,11,19,21,22} {8,9,17,18,20,23} {10,11,13,15,19,21,22} {7,12,14,16,17,18,20,23} {6,15,21,22} {9,13,14,17,18,19,23} {8,11,15,20,21,22} {10,12,17,19,23} {9,11,14,15,16,18,20,21,22} {6,17,19,23} {8,11,14,18,20,21,22} {9,17,23} {6,13,15,16,18,20,21,22} {9,14,17,19,23} {11,12,13,16,20,21,22} {10,23} {8,14,16,18,19,20,21} {10,13,15,22,23} {8,14,16,17,18,19,20,21} {5,22,23} {9,14,16,18,19,20,21} {5,17,22,23} {11,13,19,20,21} {9,12,17,18,22,23} {13,14,15,16,19,20,21} {11,18,22} {12,14,16,17,21,23} {7,18,19,20,22} {11,14,21,23} {10,15,17,18,19,20,22} {13,16,23} {11,15,19,20,22} {13,14,16,17,18,21,23} {9,19,20,22} {13,14,17,21,23} {11,20,22} {10,16,17,18,19,21,23} {14,15,20,22} {13,16,18,19,23} {12,15,17,20,21,22} {14,16,19,23} {11,17,20,21,22} {13,18,19,23} {12,21,22} {13,17,19,20,23} {15,22} {12,17,20,21,23} {15,19,22} {16,18,20,21} {13,17,19,22,23} {18,20} {15,19,21,22} {17,18,23} {15,19,20,21,22} {16,23} {17,20,22} {16,19,21,23} {20} {16,22,23} {20,21} {17,22,23} {21} {20,23} {22} {23}

Haha, even though {5} is preferred over {2,8}, the example for L(10) still has {2,8} because now it's using a successor which contains 5. This is still only an example, and neither proves nor disproves quanttheory's conjecture.
quantheory
Posts: 7
Joined: Mon Jul 07, 2014 12:31 am

### Re: Ex-Increasing Set Sequence

I've been busy this week and hadn't come back to it, but I've verified that I can replicate thundre's numbers up to n=18 using a Haskell code that assumes that the longest sequence contains all the sets containing one integer. So that's motivation to try to prove the conjecture, which is still alive and well in my view.

Here's my solution for n=18:

Code: Select all

[[1],[2],[1,4],[3],[1,2,7],[3,4],[1,2,5,7],[4],[2,3,5,7],[1,4,8],[3,6],[1,2,4,7,9],[3,5,6],[1,2,4,7,10],[5],[1,2,6,7,10],[3,4,5,9],[1,2,13],[3,4,5,6,9],[1,2,7,12],[3,4,5,6,10],[1,2,14],[4,5,6,8],[1,2,3,7,9,13],[6],[2,4,5,7,9,10],[1,3,6,8,13],[2,4,9,10],[1,3,5,6,7,8,14],[2,4,13],[1,3,6,7,8,9,11],[2,4,5,15],[1,3,6,7,8,10,11],[2,4,5,9,13],[1,3,16],[2,4,6,7,8,9,11],[1,3,5,18],[4,6,7,8,9],[1,2,3,5,10,11,16],[7],[2,4,5,8,9,10,12],[1,3,6,7,11,15],[2,4,5,8,17],[1,3,9,16],[2,4,5,6,7,10,17],[1,3,18],[2,4,5,6,7,10,11,14],[1,3,8,9,16],[2,4,5,6,7,10,18],[1,3,9,17],[2,4,5,6,7,11,18],[1,3,8,9,17],[2,4,5,6,7,11,12,14],[1,3,8,9,10,15],[2,4,5,6,7,12,18],[1,3,13,14],[2,5,6,7,8,9,10,11,12],[1,3,4,13,18],[5,6,7,8,10,11],[1,2,3,4,9,12,14,18],[8],[3,4,5,6,9,11,13,14],[1,2,7,8,10,12,17],[3,4,5,6,13,18],[1,2,10,11,17],[3,4,5,6,7,8,9,14,18],[1,2,13,17],[3,4,5,6,7,15,18],[1,2,9,10,11,17],[3,4,5,6,7,8,16,18],[1,2,10,12,17],[3,4,5,6,7,8,9,16,18],[1,2,14,17],[4,5,6,7,9,10,11,12,13],[1,2,3,8,14,15,17],[5,7,9,10,12],[1,2,3,4,6,8,11,13,14,15,18],[5,9,12],[1,2,3,6,7,8,10,11,13,17,18],[4,5,12,14],[1,2,3,7,8,9,15,16,18],[4,5,6,12,17],[1,2,3,13,16,18],[5,7,8,9,10,11,12],[1,2,3,4,6,13,16,17,18],[9],[1,2,3,4,8,14,15,17,18],[5,6,7,9,10,11,12,13],[1,2,3,8,15,17,18],[4,5,6,10,14,16],[1,2,3,7,8,9,11,12,13,17,18],[4,5,6,15,16],[1,2,3,7,8,13,14,17,18],[4,5,12,16],[1,2,3,9,15,17,18],[4,5,6,7,8,10,11,12,14,16],[1,2,3,15,17,18],[4,5,6,7,8,9,10,11,13,14,16],[1,2,12,15,17],[3,4,5,6,14,16,18],[1,2,7,8,10,12,13,15,17],[3,4,5,11,16,18],[1,2,6,8,10,13,14,15,17],[3,4,5,9,12,16,18],[1,2,13,15,17],[3,4,5,6,11,14,16,18],[1,7,8,12,13,17],[2,3,4,5,9,11,14,15,16,18],[1,7,8,10,12,13,17],[2,3,4,6,14,15,16,18],[1,7,8,9,10,11,12,13,17],[4,5,6,16,18],[1,2,7,8,9,10,12,13,14,15,17],[3,5,6,11,16,18],[1,2,7,13,14,15,17],[4,5,6,8,10,12,16,18],[1,2,7,9,11,13,14,15,17],[10],[1,2,3,6,13,15,16,17,18],[7,8,9,10,11,12,14],[1,3,6,16,17,18],[2,7,13,14,15],[1,3,4,10,11,12,16,17,18],[5,7,14,15],[1,2,6,8,9,11,12,13,16,17,18],[7,10,14],[1,4,5,6,8,11,13,15,16,17,18],[7,9,10,12,14],[2,4,5,11,16,17,18],[6,7,8,9,10,12,13,14,15],[2,5,17,18],[1,3,10,11,12,13,14,15,16],[2,7,9,17,18],[1,3,11,12,13,14,15,16],[2,8,9,10,17,18],[1,4,12,13,14,15,16],[2,6,17,18],[1,5,10,11,12,13,14,15,16],[2,8,9,17,18],[1,6,13,14,15,16],[2,7,9,11,12,17,18],[5,6,8,10,13,14,15,16],[11],[1,5,7,10,15,16,17,18],[8,9,11,12,13,14],[1,6,15,16,18],[7,8,9,10,11,12,13,14,17],[5,6,16,18],[1,7,12,13,14,15,17],[4,9,10,11,16,18],[5,7,8,12,13,14,15,17],[2,10,11,16,18],[1,8,12,13,14,15,17],[2,10,16,18],[1,9,12,13,14,15,17],[3,10,11,16,18],[5,8,9,12,13,14,15,17],[1,16,18],[2,9,12,13,14,15,17],[3,10,16,18],[2,11,14,15,17],[4,8,12,13,16,18],[7,9,10,11,14,15,17],[12],[3,8,9,14,16,17,18],[10,11,12,13,15],[1,14,16,18],[8,10,11,12,13,15,17],[3,16,18],[6,10,14,15,17],[8,9,11,12,13,16,18],[4,14,15,17],[8,10,11,12,13,16,18],[6,15,17],[5,11,12,13,14,16,18],[9,10,15,17],[6,11,13,16,18],[9,10,12,14,15,17],[13],[6,10,12,16,17,18],[11,13,14,15],[5,17,18],[9,13,14,15,16],[7,12,17,18],[10,13,14,15,16],[6,17,18],[10,14,15,16],[9,12,13,17,18],[14],[9,12,15,17,18],[13,14,16],[10,12,15,17,18],[13,16],[11,15,18],[12,14,16,17],[15],[12,14,17,18],[15,16],[12,17,18],[16],[15,18],[17],[18]]
Unfortunately, I can't test much farther yet; my code relies on my conjecture to get a little speed up, but it still has exponential scaling, and I stored the integer sets in IntSets rather than as bit patterns, so my code is probably slower than thundre's. n=18 takes about 2 minutes.
aruff
Posts: 3
Joined: Sun Jun 24, 2012 7:24 am

### Re: Ex-Increasing Set Sequence

I have not worked out all of the implications of this, but it seems really important to take into account that there are only O(n3) means.
1 <= sum <= n*(n+1)/2
1 <= count <= n
mean = sum/count
Mean has at most n2*(n+1)/2 values.

Obviously this is a weak upper bound, but the salient point is that O(n3) is a great deal better than O(2n), so if the problem can be worked in this context it might be helpful.