  Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$
elements whose permutation action results in exactly $k$ orbits or connected
components. We formulate the conjecture that, for every fixed $p$ and $n$, the
$A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a
well known property of unsigned Stirling numbers of the first kind. As the
$p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which
strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length
polynomials. In this article, we prove the $p=\infty$ case of our conjecture.
We start from an expression for the $A(p,n,k)$ which follows from an identity
by Bryan and Fulman, obtained in the their study of orbifold higher equivariant
Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The
last step essentially amounts to the log-concavity in $k$ of a generalized
Tur\'an number, namely, the maximum product of $k$ positive integers whose sum
is $n$.
