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arXiv · 1306.5708

Blocks in cycles and k-commuting permutations

Rutilo Moreno·Luis Manuel Rivera
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Abstract

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 $k \leq 4$.

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Rutilo Moreno, Luis Manuel Rivera. 2014-08-18. Blocks in cycles and k-commuting permutations. https://doi.org/10.1186/s40064-016-3638-7

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