On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions
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| Format: | Preprint |
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2024
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| author | Gajdzica, Krystian |
| author_facet | Gajdzica, Krystian |
| contents | The $A$-partition function $p_A(n)$ enumerates those partitions of $n$ whose parts belong to a fixed (finite or infinite) set $A$ of positive integers. On the other hand, the extended $A$-partition function $p_A\left(\boldsymbolμ\right)$ is defined as an multiplicative extension of the $A$-partition function to a function on $A$-partitions. In this paper, we investigate the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions. In particular, we examine the property for both the $m$-ary partition function $b_m(n)$ and the $d$-th power partition function $p_d(n)$. Moreover, we show that $b_m(\boldsymbolμ)$ ($p_d(\boldsymbolμ)$) takes its maximum value at an explicitly described set of $m$-ary partitions (power partitions), where $\boldsymbolμ$ is an $m$-ary partition (a power partition) of $n$. Additionally, we exhibit analogous results for the Fibonacci partition function and the `factorial' partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions Gajdzica, Krystian Combinatorics Number Theory Primary 05A17, 11P82, Secondary 05A20 The $A$-partition function $p_A(n)$ enumerates those partitions of $n$ whose parts belong to a fixed (finite or infinite) set $A$ of positive integers. On the other hand, the extended $A$-partition function $p_A\left(\boldsymbolμ\right)$ is defined as an multiplicative extension of the $A$-partition function to a function on $A$-partitions. In this paper, we investigate the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions. In particular, we examine the property for both the $m$-ary partition function $b_m(n)$ and the $d$-th power partition function $p_d(n)$. Moreover, we show that $b_m(\boldsymbolμ)$ ($p_d(\boldsymbolμ)$) takes its maximum value at an explicitly described set of $m$-ary partitions (power partitions), where $\boldsymbolμ$ is an $m$-ary partition (a power partition) of $n$. Additionally, we exhibit analogous results for the Fibonacci partition function and the `factorial' partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation. |
| title | On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions |
| topic | Combinatorics Number Theory Primary 05A17, 11P82, Secondary 05A20 |
| url | https://arxiv.org/abs/2401.16267 |