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

Abstract

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 $n$: $$N_{\ell }(n{)}^2-N_{\ell }(n-1)N_{\ell }(n+1)\ge 0.$$ We establish that this depends on $n(\mathrm{mod}3)$ 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 $|S_n|N_{\ell }(n)=\left|Hom\left({\mathbb{Z}}^{\ell },S_n\right)\right|$.