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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.

cesium62 wrote: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.

cesium62 wrote: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.