Computational study of non-unitary partitions
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866916385087553536 |
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| author | Akande, A. P. Genao, Tyler Haag, Summer Hendon, Maurice D. Pulagam, Neelima Schneider, Robert Sills, Andrew V. |
| author_facet | Akande, A. P. Genao, Tyler Haag, Summer Hendon, Maurice D. Pulagam, Neelima Schneider, Robert Sills, Andrew V. |
| contents | Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $ν(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures regarding the growth of $ν(n)$ as $n\to \infty$. Here we refine and prove some of these conjectures. For example, we prove $p(n) \sim ν(n)\sqrt{n/ζ(2)}$ as $n\to \infty$, and give Ramanujan-like congruences between $p(n)$ and $ν(n)$ such as $p(5n)\equiv ν(5n)\ (\operatorname{mod} 5)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_03264 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Computational study of non-unitary partitions Akande, A. P. Genao, Tyler Haag, Summer Hendon, Maurice D. Pulagam, Neelima Schneider, Robert Sills, Andrew V. Combinatorics Number Theory Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $ν(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures regarding the growth of $ν(n)$ as $n\to \infty$. Here we refine and prove some of these conjectures. For example, we prove $p(n) \sim ν(n)\sqrt{n/ζ(2)}$ as $n\to \infty$, and give Ramanujan-like congruences between $p(n)$ and $ν(n)$ such as $p(5n)\equiv ν(5n)\ (\operatorname{mod} 5)$. |
| title | Computational study of non-unitary partitions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2112.03264 |