I am in need of a way to calculate a few things about partitions. If we say there are 4 objects (n=4), then the partitions of 4 are:

4

3 + 1

2 + 2 *

2 + 1 + 1

1 + 1 + 1 + 1 *

* these partitions do not have a unique mode.

I'm looking for a function, p'(n), that will tell me how many partitions there are with unique maximum values (i.e. a unique mode). So p(4) = 5, but p'(4) = 3.

Also, I am looking for a function that can tell me what the unique maximum value is for a particular partition, i, in the set of partitions. For instance, in the partition 3+1, the unique maximum/mode is 3.

Can anyone help me with this?

Many thanks.

## Partitions with a unique maximum part

### Re: Partitions with a unique maximum part

The only definition of mode that I know is "the value that appears most often in a set of data" but that is clearly not what you mean here.

_{Jaap's Puzzle Page}

- PurpleBlu3s
**Posts:**73**Joined:**Mon Sep 19, 2011 5:49 pm

### Re: Partitions with a unique maximum part

If you think of partitioning data based on value, then what I mean is the number of occurrences of the mode.

For example, we have a set of 8 data points:

1 1 2 3 2 2 2 4

Grouping them based on their value:

(2 2 2 2) (1 1) (3) (4)

Counting the size of the groups:

4 2 1 1

This is a partition of 8, and the number of occurrences of the mode is 4.

For example, we have a set of 8 data points:

1 1 2 3 2 2 2 4

Grouping them based on their value:

(2 2 2 2) (1 1) (3) (4)

Counting the size of the groups:

4 2 1 1

This is a partition of 8, and the number of occurrences of the mode is 4.

### Re: Partitions with a unique maximum part

The classic way of calculating the partitions is recursive using the P

https://en.wikipedia.org/wiki/Partition ... r_of_parts

So you can use these results and calc p*(n) = ∑

_{k}(n) = Partitions of n with k as maximum:https://en.wikipedia.org/wiki/Partition ... r_of_parts

So you can use these results and calc p*(n) = ∑

_{i=2}^{n}P_{i-1}(n-i)-
**Posts:**4**Joined:**Sun Aug 28, 2016 3:27 pm

### Re: Partitions with a unique maximum part

I know this is old, but I just saw this on reviewing the forums and I wanted to share my thoughts (spoilered for anyone who wants to work on it).

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