On a mod 3 property of \(\ell \)-tuples of pairwise commuting permutations

Let \(S_n\) $S_n$ denote the symmetric group of permutations acting on n elements. We investigate the double sequence \(\{N_{\ell }(n)\}\) $\{N_{\ell }(n)\}$ counting the number of \(\ell \) $\ell$ tuples of elements of the symmetric group \(S_n\) $S_n$ , where the components commute, normalized by the order of \(S_n\) $S_n$ . Our focus lies on exploring log-concavity with respect to n: \(\begin{aligned} N_{\ell }(n)^2 - N_{\ell }(n-1) \,\, N_{\ell }(n+1) \ge 0. \end{aligned}\) $$\begin{array}{c}\hfill N_{\ell }{(n)}^2-N_{\ell }(n-1)N_{\ell }(n+1)\ge 0.\end{array}$$ We establish that this depends on \(n \pmod {3}\) $n(mod3)$ for sufficiently large \(\ell \) $\ell$ . These numbers are studied by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, \(N_2(n)\) $N_2(n)$ represents the partition numbers p(n), while \(N_{3}(n)\) $N_3(n)$ represents the number of non-equivalent n-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since \( \vert S_n \vert \,\, N_{\ell }(n) = \left| \textrm{Hom}\left( \mathbb {Z}^{\ell },S_n\right) \right| \) $|S_n|N_{\ell }(n)=\left|\text{Hom},\left({\mathbb{Z}}^{\ell },,,S_n\right)\right|$ .