Bessenrodt--Ono inequalities for $\ell$-tuples of pairwise commuting permutations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917771234770944 |
|---|---|
| author | Abdesselam, Abdelmalek Heim, Bernhard Neuhauser, Markus |
| author_facet | Abdesselam, Abdelmalek Heim, Bernhard Neuhauser, Markus |
| contents | 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 $\ell$-tuples $π=\left( π_1, \ldots, π_{\ell}\right) \in S_n^{\ell}$ with $π_i π_j = π_j π_i$ for $1 \leq i,j \leq \ell$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b, \ell \geq 2$. 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 $\ell =2$ and $a+b >9$. In this paper we prove that for each pair $(a,b)$ the sign of $\{Δ_{a,b}^{\ell} \}_{\ell}$ 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 $n$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bessenrodt--Ono inequalities for $\ell$-tuples of pairwise commuting permutations Abdesselam, Abdelmalek Heim, Bernhard Neuhauser, Markus Combinatorics Number Theory 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 $\ell$-tuples $π=\left( π_1, \ldots, π_{\ell}\right) \in S_n^{\ell}$ with $π_i π_j = π_j π_i$ for $1 \leq i,j \leq \ell$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b, \ell \geq 2$. 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 $\ell =2$ and $a+b >9$. In this paper we prove that for each pair $(a,b)$ the sign of $\{Δ_{a,b}^{\ell} \}_{\ell}$ 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 $n$-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. |
| title | Bessenrodt--Ono inequalities for $\ell$-tuples of pairwise commuting permutations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2409.04881 |