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

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