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

2024/03/03 by Abdesselam, Abdelmalek, Heim, Bernhard, Neuhauser, Markus · 1 citation
#05A17 #11P82 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.01441

Abstract

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. We establish that this depends on n \pmod3 for sufficiently large ℓ. 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 \vert S_n \vert N_ℓ(n) = \vert Hom ( ℤ^ℓ,S_n) \vert .

Cited by

• Log-Concavity and Log-Convexity of Restricted Infinite Products

Related

• Asymptotics of commuting ℓ-tuples in symmetric groups and log-concavity
• Bessenrodt--Ono inequalities for ℓ-tuples of pairwise commuting permutations
• A bijection for tuples of commuting permutations and a log-concavity conjecture
• Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups
• Symmetric products, permutation orbifolds and discrete torsion