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

Let Sₙ denote the symmetric group of permutations acting on n elements. We investigate the double sequence _(n)\ counting the number of tuples of elements of the symmetric group Sₙ, where the components commute, normalized by the order of Sₙ. Our focus lies on exploring log-concavity with respect to n: N_(n)² - N_(n-1) \,\, N_(n+1) ≥ 0. We establish that this depends on n 3 for sufficiently large . These numbers are studied by Bryan and Fulman as the nth orbifold characteristics, generalizing work of Macdonald and Hirzebruch--Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, N₂(n) represents the partition numbers $p(n)$, while N₃(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 Sₙ \,\, N_(n) = Hom ( Z^,Sₙ).