Blocks in cycles and k-commuting permutations

TL;DR

This paper characterizes permutations that $k$-commute with a given permutation using cycle block structures, providing formulas for specific cases and counts for small $k$, advancing understanding of permutation commutation properties.

Contribution

It introduces a cycle block-based characterization of $k$-commuting permutations and derives explicit formulas for their counts in key cases.

Findings

Characterization of $k$-commuting permutations via cycle blocks

Formulas for permutations that $k$-commute with transpositions and involutions

Counts of $k$-commuting permutations for $k extless= 4

Abstract

Let $k$ be a nonnegative integer, and let $\alpha$ and $\beta$ be two permutations of $n$ symbols. We say that $\alpha$ and $\beta$ $k$-commute if $H(\alpha\beta, \beta\alpha)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $\beta$ in terms of blocks in cycles in the decomposition of $\beta$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute with a transposition, a fixed-point free involution and an $n$-cycle, for any $k$. Also, we determine the number of permutations that $k$-commute with any given permutation, for $k \leq 4$.