show that the number of standard tableau of shape $(n^2)$ is the Catalan number

Question

How would one show that the number of standard tableau of shape $(n^2)$ is the Catalan number

$\mathrm{\frac{1}{n+1}}$$2n\choose{n}$

any help would be great.

Answer 1

I’m going to assume that by shape $(n^2)$ you mean $2\times n$ tableaux, as the result is not true for $n\times n$ tableaux. First verify that there is a bijection between standard tableaux of this shape and Dyck paths [PDF] of length $2n$: the numbers in the first row of the tableau are the positions of the up-steps in the Dyck path, and the numbers in the second row are the positions of the down-steps. Then use or prove the result that the number of Dyck paths (or Dyck words, or balanced parenthesis strings) of length $2n$ is $C_n$.

You can also use the hook length formula to get the result.