A bijection for tuples of commuting permutations and a log-concavity conjecture

Let A(ℓ,n,k) denote the number of ℓ-tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We provide a new proof of an explicit formula for A(ℓ,n,k) which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the ℓ+1→ℓ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of A(ℓ,n,k) with respect to k. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula.

Recommended citation: Abdesselam, A., Brunialti, P., Doan, T., & Velie, P. (2024). A bijection for tuples of commuting permutations and a log-concavity conjecture. Research in Number Theory, 10(2), 45.
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