All possible ways to order numbers in an array with decreasing rows and columns Given positive integer numbers $1,2,...,N\cdot M$. How many ways are there to order them in an $N\times M$ array given that the values decrease in each row from left to right and in each column from top to bottom? For small arrays one can just count but I don't find a general rule. Thanks for any help.

This is the number of standard Young tableaux for a Young diagram with $N$ rows and $M$ columns. By the hook length formula, this is
$$
\frac{(NM)!}{\prod_{i=1}^M\prod_{j=1}^N(i+j-1)}\;.
$$
This is OEIS sequence A060854. That entry gives the alternative formula
$$
(NM)!\prod_{k=0}^{N-1}\frac{k!}{(M+k)!}\;.
$$
