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Abstract 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.
Abdesselam et al. (Wed,) studied this question.