## Permuted Matrices

Published on Saturday, 7th May 2016, 04:00 pm; Solved by 128;
Difficulty rating: 100%

### Problem 559

An ascent of a column j in a matrix occurs if the value of column j is smaller than the value of column j+1 in all rows.

Let P(k, r, n) be the number of r x n matrices with the following properties:

• The rows are permutations of {1, 2, 3, ... , n}.
• Numbering the first column as 1, a column ascent occurs at column j<n if and only if j is not a multiple of k.

For example, P(1, 2, 3) = 19, P(2, 4, 6) = 65508751 and P(7, 5, 30) mod 1000000123 = 161858102.

Let Q(n) =$\, \displaystyle \sum_{k=1}^n\,$ P(k, n, n).
For example, Q(5) = 21879393751 and Q(50) mod 1000000123 = 819573537.

Find Q(50000) mod 1000000123.