Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations

TL;DR

This paper investigates the log-concavity of the number of commuting permutation tuples with a fixed number of orbits, proving the conjecture for the case of infinitely many permutations using asymptotic analysis and Turán numbers.

Contribution

It proves the log-concavity conjecture for the case of infinitely many commuting permutations, extending known results and connecting to Turán numbers.

Findings

Proves the $p= fty$ case of the log-concavity conjecture.

Derives asymptotics for $A(p,n,k)$ as $p o abla$.

Establishes a link between permutation orbit counts and Turán numbers.

Abstract

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the $A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a well known property of unsigned Stirling numbers of the first kind. As the $p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length polynomials. In this article, we prove the $p=\infty$ case of our conjecture. We start from an expression for the $A(p,n,k)$ which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The last step essentially amounts to the log-concavity in $k$ of a generalized Tur\'an number, namely, the maximum product of $k$ positive integers whose sum is $n$.