A combinatorial approach to the power of 2 in the number of involutions

TL;DR

This paper introduces a combinatorial method to analyze the highest power of 2 dividing the count of involutions and related permutations, revealing new insights into their algebraic and combinatorial properties.

Contribution

It presents a novel combinatorial approach to determine the largest power of 2 dividing involution counts and related permutation sums.

Findings

Identifies the maximum power of 2 dividing involution counts

Analyzes the signed sum of involutions and parity-based involution counts

Provides combinatorial formulas for these divisibility properties

Abstract

We provide a combinatorial approach to the largest power of $p$ in the number of permutations $\pi$ with $\pi^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.