Counting Permutations in $S_{2n}$ and $S_{2n+1}$

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.