Log-Concavity with Respect to the Number of Orbits for Infinite Tuples of Commuting Permutations

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.