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_{ (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).