Bessenrodt--Ono inequalities for ℓ-tuples of pairwise commuting permutations

2024/09/07 by Abdelmalek Abdesselam, Abdesselam, Abdelmalek, Bernhard Heim +3
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.04881

openalex publication_date 2024/09/07 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

Let S_n denote the symmetric group. We consider N_ℓ(n) := \frac\vert Hom( ℤ^ℓ,S_n) \vertn! which also counts the number of ℓ-tuples π=( π₁, …, π_ℓ) ∈ S_n^ℓ with π_i π_j = π_j π_i for 1 ≤ i,j ≤ ℓ scaled by n!. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let a,b, ℓ ≥ 2. It is known by Bringman, Franke, and Heim, that the Bessenrodt--Ono inequality Δ_a,b^ℓ:= N_ℓ(a) N_ℓ(b) - N_ℓ(a+b) gt;0 is valid for a,b ≫ 1 and by Bessenrodt and Ono that it is valid for ℓ =2 and a+b >9. In this paper we prove that for each pair (a,b) the sign of \Δ_a,b^ℓ \_ℓ is getting stable. In each case we provide an explicit bound. The numbers N_ℓ( n) had been identified by Bryan and Fulman as the n-th orbifold characteristics, generalizing work by Macdonald and Hirzebruch--Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where N₂(n)=p(n) represents the partition function.

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