It not quite clear what you mean by allowing repeated numbers, but what one usually considers in that case is so-called semi-standard Young tableaux, i.e., tableaux which are increasing (strict inequality) down each column, but only nondecreasing (equality allowed) along each row. The number of such arrangements on a given Young diagram, where the numbers $1,2,\dots,N$ are allowed, is counted as follows: define the "content" of box $(i,j)$ in the diagram to be $i-j$. Here's an illustration:
0 1 2 3 4 -1 0 1 2 -2 -1 0 -3 -2 -4 -3 -5
Hook lengths are defined as for the usual hook-length formula for counting standard Young tableaux:
10 8 5 3 1 8 6 3 1 6 4 1 4 2 3 1 1
To get the answer, take the product over all boxes of (($N$ plus the content of that box) divided by (the hook length for that box)).
(This is a special case of something called Stanley's hook-content formula.)