Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913373328769024 |
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| author | Ballantine, Cristina Welch, Amanda |
| author_facet | Ballantine, Cristina Welch, Amanda |
| contents | We consider the number of various partitions of $n$ with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for $n\geq 5$ we have $p_{od}^{eu}(n)<p_{ed}^{ou}(n)$, where $p_{od}^{eu}(n)$ is the number of partitions of $n$ with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and $p_{ed}^{ou}(n)$ is the number of partitions of $n$ with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity Ballantine, Cristina Welch, Amanda Combinatorics Number Theory 05A17, 05A20, 11P81 We consider the number of various partitions of $n$ with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for $n\geq 5$ we have $p_{od}^{eu}(n)<p_{ed}^{ou}(n)$, where $p_{od}^{eu}(n)$ is the number of partitions of $n$ with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and $p_{ed}^{ou}(n)$ is the number of partitions of $n$ with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts. |
| title | Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity |
| topic | Combinatorics Number Theory 05A17, 05A20, 11P81 |
| url | https://arxiv.org/abs/2406.00139 |