Tags: #theorem

Statement

The number of standard Young tableauxs of shape $(n,n)$ is equal to $C_n$, the Catalan numbers.

Proof

We shall construct a bijection between these and Dyck paths.
The numbers in the first row will tell you where the up steps are
The numbers in the second row will tell you where the down steps are.

This gives us a valid Dyck path as each subtableaux (ie. 1 through d < n) is a valid tableaux, so there are at least as many in the first row than the second row. Thus, we never go down more than we go up.