On products of permutations with the most uncontaminated cycles by designated labels

• Published in: arXiv.org
• Main Author: Chen, Ricky X F
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 31.10.2022
• Subjects: Labels; Mathematics - Combinatorics; Mathematics - Group Theory; Permutations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2210.17080

There is a growing interest in studying the distribution of certain labels in products of permutations since the work of Stanley addressing a conjecture of B\'{o}na. This paper is concerned with a problem in that direction. Let \(D\) be a permutation on the set \([n]=\{1,2,\ldots, n\}\) and \(E\subset [n]\). Suppose the maximum possible number of cycles uncontaminated by the \(E\)-labels in the product of \(D\) and a cyclic permutation on \([n]\) is \(\theta\) (depending on \(D\) and \(E\)). We prove that for arbitrary \(D\) and \(E\) with few exceptions, the number of cyclic permutations \(\gamma\) such that \(D\circ \gamma\) has exactly \(\theta-1\) \(E\)-label free cycles is at least \(1/2\) that of \(\gamma\) for \(D\circ \gamma\) to have \(\theta\) \(E\)-label free cycles, where \(1/2\) is best possible. An even more general result is also conjectured.