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

2023/09/14 by Abdelmalek Abdesselam, Abdesselam, Abdelmalek · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2309.07358

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=∞ 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→∞ 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.

Cited by

• Proof of a conjecture by Starr and log-concavity for random commuting permutations
• A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations
• Some Observations about the "Generalized Abundancy Index"

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