# how many (5,4,3,2,1)-tables are there?

A (5,4,3,2,1)-table is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is (strictly) increasing. How many (5,4,3,2,1)-tables are there?

I'm not sure how to approach this question. A brute force computer program that checks all 15! permutations yields an answer of 292864. Obviously the configuration is uniquely specified once the entries of each row are chosen and the main difficulty is that not only each row, but also each column, must be increasing. It might be that some numbers must take on certain positions. For instance, 15 must clearly be the last entry in its row and column. The only way 14 would not be the last entry in its row is if 15 is the last entry in its row, but 14 has to be the last entry in either its row or column. The number 1 must be in the first row and first column. This reasoning unfortunately doesn't scale well. Perhaps considering (2,1)- or (3,2,1)- tables (defined in the natural way) first might make some patterns clearer? There are 2 (2,1)-tables.