problem 559

[cesium62]

When I count P(1, 2, 3) I get 14 and not 19 as given in problem Problem 559 (View Problem). There are 36 2x3 matrices associated with the problem, right? I wrote down the 36 matrices. For each, under the first column, I write an 'a' if row 1 column 1 is smaller than row 2 column 1. Similarly under the 2nd column I write an 'a' if row 1 column 2 is smaller than row 2 column 2. I ignore the 3rd column. I count the number of matrices that do not have an 'a' underneath. I keep counting 14 and not 19.

I'm not going to figure out the answer in any event, but I wonder if anyone could help me understand what part of the problem I'm not reading correctly.

[jaap]

For each, under the first column, I write an 'a' if row 1 column 1 is smaller than row 2 column 1. Similarly under the 2nd column I write an 'a' if row 1 column 2 is smaller than row 2 column 2. I ignore the 3rd column.

An ascent occurs when a whole column is smaller than the column to its right. So the above should read:

For each, under the first column, I write an 'a' if row 1 column 1 is smaller than row 1 column 2, and row 2 column 1 is smaller than row 2 column 2. Similarly under the 2nd column I write an 'a' if row 1 column 2 is smaller than row 1 column 3, and row 2 column 2 is smaller than row 2 column 3.

[Animus]

For each, under the first column, I write an 'a' if row 1 column 1 is smaller than row 2 column 1. Similarly under the 2nd column I write an 'a' if row 1 column 2 is smaller than row 2 column 2. I ignore the 3rd column. I count the number of matrices that do not have an 'a' underneath. I keep counting 14 and not 19.

Hi, cesium62,
jaap is rigth, your mixing up rows and columns. A column ascent does not mean that the elements in any column appear in ascending order, but that for the column examined the element in this column is smaller than the element in the next column for all rows.
Thus, for example, $\begin{pmatrix} 2 & 1 & 3 \\ 3 & 2 & 1 \end{pmatrix}$ shows no column ascent and counts for P(1,2,3) although the elements in column 1 and column 2 are both increasing.

[cesium62]

Thanks.

[schang1146]

Out of the 36 possible matrices, I still only count 17:

Expand

123
123

123
132

123
213

123
312

132
123

132
132

132
231

213
123

213
213

213
312

231
123

231
132

231
231

312
123

312
213

312
312

What two am I missing or am I not understanding the question right?

[mdean]

Out of the 36 possible matrices, I still only count 17:

What two am I missing or am I not understanding the question right?

By the look of things, it's quite possible you've listed every matrix not included in P(1,2,3).