Toward resolving Kang and Park's generalization of the Alder-Andrews Theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916337048092672 |
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| author | Sturman, Leah Swisher, Holly |
| author_facet | Sturman, Leah Swisher, Holly |
| contents | The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in 2008, and Alfes, Jameson, and Lemke Oliver in 2010. While Andrews and Yee utilized $q$-series and combinatorial methods, Alfes et al. proved the finite number of remaining cases using asymptotics originating with Meinardus together with high-performance computing. In 2020, Kang and Park conjectured a "level $2$" Alder-Andrews type partition inequality which relates to the second Rogers-Ramanujan identity. Duncan, Khunger, the second author, and Tamura proved Kang and Park's conjecture for all but finitely many cases using a combinatorial shift identity. Here, we generalize the methods of Alfes et al. to resolve nearly all of the remaining cases of Kang and Park's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward resolving Kang and Park's generalization of the Alder-Andrews Theorem Sturman, Leah Swisher, Holly Number Theory Combinatorics 11P82, 11P84, 11P81 The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in 2008, and Alfes, Jameson, and Lemke Oliver in 2010. While Andrews and Yee utilized $q$-series and combinatorial methods, Alfes et al. proved the finite number of remaining cases using asymptotics originating with Meinardus together with high-performance computing. In 2020, Kang and Park conjectured a "level $2$" Alder-Andrews type partition inequality which relates to the second Rogers-Ramanujan identity. Duncan, Khunger, the second author, and Tamura proved Kang and Park's conjecture for all but finitely many cases using a combinatorial shift identity. Here, we generalize the methods of Alfes et al. to resolve nearly all of the remaining cases of Kang and Park's conjecture. |
| title | Toward resolving Kang and Park's generalization of the Alder-Andrews Theorem |
| topic | Number Theory Combinatorics 11P82, 11P84, 11P81 |
| url | https://arxiv.org/abs/2407.18350 |