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

Ponente(s): Markus Neuhauser Reiter, Abdelmalek Abdesselam; Bernhard Heim; Markus Neuhauser

Let $S_n$ denote the symmetric group. We consider \begin{equation*} N_l(n):=\frac{|Hom(\mathbb{Z}^l,S_n)|}{n!} \end{equation*} which also counts the number of $l$-tuples $\pi=( \pi_1, \ldots, \pi_l\right) \in S_n^l$ with $\pi_i\pi_j=\pi_j\pi_i$ for $1\leq i,j \leq l$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryan and Fulman, and White. Let $a,b,\ell \geq 2$. It is known by Bringmann, Franke, and Heim, that the Bessenrodt-Ono inequality \begin{equation*} \Delta_{a,b}^l:=N_l(a)N_l(b)-N_l(a+b)>0 \end{equation*} is valid for $a,b\gg 1$ and by Bessenrodt and Ono that it is valid for $l=2$ and $a+b >9$. In this presentation we will show that for each pair $(a,b)$ the sign of $\{\Delta_{a,b}^l \}_l$ 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-H\"ofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where $N_2(n)=p(n)$ represents the partition function.