  Let $\alpha(n)$ denote the number of perfect square permutations in the
symmetric group $S_n$. The conjecture $\alpha(2n+1) = (2n+1) \alpha(2n)$,
provided by Stanley[4], was proved by Blum[1] using a generating function. This
paper presents a combinatorial proof for this conjecture. At the same time, we
demonstrate that all permutations with an even number of even cycles in both
$S_{2n}$ and $S_{2n+1}$ can be categorized into three distinct types that
correspond to each other.
