Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912212239515648 |
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| author | Schlosser, Michael J. Zhou, Nian Hong |
| author_facet | Schlosser, Michael J. Zhou, Nian Hong |
| contents | We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$), as $n$ tends to infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_14365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions Schlosser, Michael J. Zhou, Nian Hong Combinatorics Classical Analysis and ODEs Number Theory 05A17 (Primary) 11P81 (Secondary) We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$), as $n$ tends to infinity. |
| title | Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions |
| topic | Combinatorics Classical Analysis and ODEs Number Theory 05A17 (Primary) 11P81 (Secondary) |
| url | https://arxiv.org/abs/2408.14365 |