  Let $S_n$ denote the symmetric group of permutations acting on $n$ elements.
We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of
$\ell$ tuples of elements of the symmetric group $S_n$, where the components
commute, normalized by the order of $S_n$. Our focus lies on exploring
log-concavity with respect to $n$: $$ N_{\ell}(n)^2 - N_{\ell}(n-1) \,\,
N_{\ell}(n+1) \geq 0.$$ We establish that this depends on $n \pmod{3}$ for
sufficiently large $\ell$. These numbers are studied by Bryan and Fulman as the
$n$th orbifold characteristics, generalizing work of Macdonald and
Hirzebruch--Hofer concerning the ordinary and string-theoretic Euler
characteristics of symmetric products. Notably, $N_2(n)$ represents the
partition numbers $p(n)$, while $N_{3}(n)$ represents the number of
non-equivalent $n$-sheeted coverings of a torus studied by Liskovets and
Medynkh. The numbers also appear in algebra since $ \vert S_n \vert \,\,
N_{\ell}(n) = \left\vert Hom \left( \mathbb{Z}^{\ell},S_n\right) \right\vert $.
