Integer sequences and k-commuting permutations

TL;DR

This paper derives formulas for counting permutations that k-commute with a given permutation, relating these counts to integer sequences and providing new interpretations for some sequences in the OEIS database.

Contribution

It introduces formulas for c(k, β) for specific cycle types and connects these formulas to OEIS sequences, offering new insights and interpretations.

Findings

Formulas for c(k, β) for certain cycle types

Connections between permutation counts and OEIS sequences

New interpretations for some integer sequences in OEIS

Abstract

Let $\beta$ be any permutation on $n$ symbols and let $c(k, \beta)$ be the number of permutations that $k$-commute with $\beta$. The cycle type of a permutation $\beta$ is a vector $(c_1, \dots, c_n)$ such that $\beta$ has exactly $c_i$ cycles of length $i$ in its disjoint cycle factorization. In this article we obtain formulas for $c(k, \beta)$, for some cycle types. We also express these formulas in terms of integer sequences as given in "The On-line Encyclopedia of Integer Sequences" (OEIS). For some of these sequences we obtain either new interpretations or relationships with sequences in the OEIS database.