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