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

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