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

TL;DR

This paper provides a combinatorial proof for a conjecture relating perfect square permutations in symmetric groups and classifies permutations with even cycles into three types.

Contribution

It offers a combinatorial proof of Stanley's conjecture and classifies permutations with even cycles in symmetric groups into three corresponding types.

Findings

Proved Stanley's conjecture combinatorially.

Classified permutations with even cycles into three types.

Established correspondence between permutation types.

Abstract

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.