Inequalities for $k$-regular partitions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913393292607488 |
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| author | Heim, Bernhard Neuhauser, Markus |
| author_facet | Heim, Bernhard Neuhauser, Markus |
| contents | We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the $k$-regular partition functions $p_k(n)$. Our focus is on addressing the solutions of the Bessenrodt--Ono inequality \begin{equation*} p_k(a) \, p_k(b) > p_k(a+b). \end{equation*} We determine the sets $E_k$ and $F_k$ consisting of all pairs $(a,b)$, where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets $E_{\infty}$ and $F_{\infty}$ for the partition function $p(n)$. We prove by induction that $E_k=E_{\infty}$ and $F_k=F_{\infty}$ if and only if $k \geq 10$. Beckwith and Bessenrodt used analytic methods to consider $2 \leq k \leq 6$, while Alanazi, Gagola, and Munagi studied the case $k=2$ using combinatorial methods. Finally, we present a precise and comprehensive conjecture on the log-concavity of the $k$-regular partition function extending previous speculations by Craig and Pun. The case $k=2$ was recently proven by Dong and Ji. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inequalities for $k$-regular partitions Heim, Bernhard Neuhauser, Markus Combinatorics We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the $k$-regular partition functions $p_k(n)$. Our focus is on addressing the solutions of the Bessenrodt--Ono inequality \begin{equation*} p_k(a) \, p_k(b) > p_k(a+b). \end{equation*} We determine the sets $E_k$ and $F_k$ consisting of all pairs $(a,b)$, where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets $E_{\infty}$ and $F_{\infty}$ for the partition function $p(n)$. We prove by induction that $E_k=E_{\infty}$ and $F_k=F_{\infty}$ if and only if $k \geq 10$. Beckwith and Bessenrodt used analytic methods to consider $2 \leq k \leq 6$, while Alanazi, Gagola, and Munagi studied the case $k=2$ using combinatorial methods. Finally, we present a precise and comprehensive conjecture on the log-concavity of the $k$-regular partition function extending previous speculations by Craig and Pun. The case $k=2$ was recently proven by Dong and Ji. |
| title | Inequalities for $k$-regular partitions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.10987 |