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

TL;DR

This paper investigates the distribution of uncontaminated cycles in permutation products, establishing bounds on the number of permutations with nearly maximum uncontaminated cycles, extending prior work on permutation label distributions.

Contribution

It proves bounds on the proportion of permutations with one fewer uncontaminated cycle than the maximum, generalizing previous results and conjecturing further extensions.

Findings

At least half as many permutations have heta-1 uncontaminated cycles as those with heta.

The result holds for most permutations with few exceptions.

A more general conjecture is proposed for future research.

Abstract

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.