Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations
Abdelmalek Abdesselam
TL;DR
The paper investigates the log-concavity of the counts $A(p,n,k)$ of ordered commuting $p$-tuples of permutations of $[n]$ by the number of orbits $k$. It proves the conjectured log-concavity in the asymptotic regime $p\to\infty$ by deriving precise asymptotics $A(p,n,k)\sim F(n,k)\,E(n,k)^p$, where $E(n,k)$ is the maximal product of $k$ positive integers summing to $n$ and $F(n,k)$ is a computable multiplicative factor; crucially, $E(n,k)$ satisfies the log-concavity $E(n,k)^2\ge E(n,k-1)E(n,k+1)$. The analysis hinges on an explicit formula for $A(p,n,k)$ in terms of a function $B(p,n)$ and on identifying dominant terms in a sum over compositions, linking the growth rate to a generalized Turán number via $E(n,k)$. By combining these asymptotics with a careful case analysis of the Euclidean-divisions-derived parameters, the authors establish the $p=\infty$ case of the conjecture, providing evidence for the full log-concavity conjecture and revealing deep connections between permutation-group enumeration and extremal graph theory.
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 Turán number, namely, the maximum product of $k$ positive integers whose sum is $n$.