On a mod 3 property of ℓ-tuples of pairwise commuting permutations: On a mod 3 property of ℓ-tuples...: A. Abdesselam et al.

Let S n denote the symmetric group of permutations acting on n elements. We investigate the double sequence { N ℓ (n) } counting the number of ℓ 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 ℓ (n) 2 - N ℓ (n - 1) N ℓ (n + 1) ≥ 0. <graphic mime-subtype="GIF" href="11139_2024_967_Article_Equ2.gif"></graphic> We establish that this depends on n (mod 3) for sufficiently large ℓ . 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) 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 ℓ (n) = Hom Z ℓ , S n . [ABSTRACT FROM AUTHOR]