Finite analogs of partition bias related to hook length two and a variant of Sylvester's map
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916685004406784 |
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| author | Berkovich, Alexander Dhar, Aritram |
| author_facet | Berkovich, Alexander Dhar, Aritram |
| contents | In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of $n$ in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite analogs of partition bias related to hook length two and a variant of Sylvester's map Berkovich, Alexander Dhar, Aritram Combinatorics Number Theory 05A15, 05A17, 05A19, 11P81, 11P82 In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of $n$ in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part. |
| title | Finite analogs of partition bias related to hook length two and a variant of Sylvester's map |
| topic | Combinatorics Number Theory 05A15, 05A17, 05A19, 11P81, 11P82 |
| url | https://arxiv.org/abs/2503.17571 |