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

Let A( ,n,k) 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( ,n,k) 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 +1→ ℓ+1→ℓ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of A( ,n,k) A(ℓ,n,k) with respect to k . The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula.