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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)`.