On log-concavity of the number of orbits in commuting tuples of permutations

TL;DR

This paper investigates the log-concavity of the number of commuting permutation tuples with a fixed number of orbits, proving it for large n when the number of orbits is close to n, extending previous results.

Contribution

It establishes the log-concavity of A(p, n, k) for large n when k is near n, generalizing prior findings to a broader range of p and k.

Findings

Proves log-concavity of A(p, n, k) for large n when k=n−α.

Extends log-concavity results beyond the p=∞ case.

Supports the conjecture for a wider parameter range.

Abstract

Denote by $A(p, n, k)$ the number of commuting $p$-tuples of permutations on $[n]$ that have exactly $k$ distinct orbits. It was conjectured in~\cite{abdesselam2023log} that $A(p, n, k)$ is log-concave with respect to $k$ for every $p\geq 2, n\geq 3$, and the log-concavity was proved in ``$p=\infty$" case. In this paper, we prove that for $k=n-\alpha$, the log-concavity for $A(p, n, k)$ holds for every $p\geq 2$ for sufficiently large $n$.