## Exercise P

Posted 03/26/2015
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Suppose T is a Young tableau of shape \lambda=n^2. If we replace the k-th largest element in T by k and rotate T 180 degrees, then we get a new tableau, say P. If P=T, we say T is symmetric. As examples, in the following picture, T_1 is a symmetric Young tableau and T_2 is not symmetric.

a(n) is the number of symmetric standard Young tableaux of shape (n^2).

For example, a(1)=1, a(2)=1, a(3)=3.

Find a(n).