Blocks in cycles and k-commuting permutations
Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, 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 $β$ in terms of blocks in cycles in the decomposition of $β$ 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 $k \leq 4$.