Bessenrodt–Ono Inequalities for \(\ell \)-Tuples of Pairwise Commuting Permutations

Abstract

Let \(S_n\) denote the symmetric group. We consider

$$\begin{aligned} N_{\ell }(n):= \frac{\left| \textrm{Hom}\left( {\mathbb {Z}}^{\ell },S_n\right) \right| }{n!} \end{aligned}$$

which also counts the number of \(\ell \)-tuples \(\pi =\left( \pi _1, \ldots , \pi _{\ell }\right) \in S_n^{\ell }\) with \(\pi _i \pi _j = \pi _j \pi _i\) for \(1 \le i,j \le \ell \) scaled by n!. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryan and Fulman, and White. Let \(a,b, \ell \ge 2.\) It is known by Bringmann, Franke, and Heim, that the Bessenrodt–Ono inequality

$$\begin{aligned} \Delta _{a,b}^{\ell }:= N_{\ell }(a) \, N_{\ell }(b) - N_{\ell }(a+b) >0 \end{aligned}$$

is valid for \(a,b \gg 1\) and by Bessenrodt and Ono that it is valid for \(\ell =2\) and \(a+b >9.\) In this paper, we prove that for each pair (a, b) the sign of \(\{\Delta _{a,b}^{\ell } \}_{\ell }\) is getting stable. In each case we provide an explicit bound. The numbers \(N_{\ell }\left( n\right) \) had been identified by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where \(N_2(n)=p(n) \) represents the partition function.