Bessenrodt--Ono inequalities for $\mathrm{\ell }$-tuples of pairwise commuting permutations

Let $S_n$ denote the symmetric group. We consider \begin{equation*} N_{\ell}(n) := \frac{\left\vert Hom\left( $\mathbb{Z}$^{\ell},S_n\right) \right\vert}{n!} \end{equation*} which also counts the number of $\mathrm{\ell }$-tuples $\pi =({\pi }_1,\dots ,{\pi }_{\mathrm{\ell }})\in S_n^{\mathrm{\ell }}$ with ${\pi }_i{\pi }_j={\pi }_j{\pi }_i$ for $1\le i,j\le \mathrm{\ell }$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b,\mathrm{\ell }\ge 2$. It is known by Bringman, Franke, and Heim, that the Bessenrodt–Ono inequality \begin{equation*} Δ_{a,b}^{\ell}:= N_{\ell}(a) , N_{\ell}(b) - N_{\ell}(a+b) >0 \end{equation*} is valid for $a,b\gg 1$ and by Bessenrodt and Ono that it is valid for $\mathrm{\ell }=2$ and $a+b>9$. In this paper we prove that for each pair $(a,b)$ the sign of $\{{\mathrm{\Delta }}_{a,b}^{\mathrm{\ell }}{\}}_{\mathrm{\ell }}$ is getting stable. In each case we provide an explicit bound. The numbers $N_{\mathrm{\ell }}\left(n\right)$ 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 $N_2(n)=p(n)$ represents the partition function.