Count of Matrices

[davar55]

Call an m x n matrix of real numbers "ascending" if
every row and column is in ascending (increasing) order,
i.e. a_i+1,j > a_i,j and a_i,j+1 > a_i,j.

Call an m x n matrix "sequential" if its m*n entries consist of
all the integers 1 through mn inclusive.

How many ascending sequential m x n matrices are there?

(I think this is a hard problem. I think I have a partial answer.)

[Kevin]

http://mathworld.wolfram.com/YoungTableau.html

http://mathworld.wolfram.com/HookLengthFormula.html

I think that's the answer you're looking for. The definition given for hook length formula doesn't explicitly mention it, but the formula tells you the number of standard Young tableau there are for a given Young diagram. Your question essentially asks "How many standard Young tableaux are there corresponding to the Young diagram that is an mxn rectangle?"

I'd suggest you keep working for a while before you look at the answer and proofs of it. I don't know for sure, but on the surface it at least seems doable if you're sufficiently interested in the question/subject to put some time and thought into it.

[davar55]

That general formula gives (mn)! / (product of hook lengths),
which for an m x n rectangle is in a relatively simple form.
In particular for n x 2 (or 2 x n) rectangles, the formula gives
(2n)! / (n! * (n+1)!), i.e. the Catalan numbers 1,2,5,14,42,etc.
This much I had confirmed empirically.