From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ rectangle are there?
Also, what if these numbers can be repeated?
Many thanks.
Best Answer
For the answer to your main question, you need to use the hook-length formula.
OEIS A060854 gives the result $$(mn)! \prod_{i=0}^{n-1} \frac{i!}{(m+i)!} \textrm{ or equivalently } (mn)! \prod_{j=0}^{m-1} \frac{j!}{(n+j)!} $$ and some more information.