On the $k$th smallest part of a partition into distinct parts
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913237507768320 |
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| author | Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph |
| author_facet | Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph |
| contents | A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $π$ into distinct parts of any integer $n$, namely $s_k(π)$. Using $s_k(π)$, we generalize the above result for the $k$th smallest parts of partitions for any positive integer $k$ and show its connection with divisor functions for general $k$ and derive interesting special cases. We also study weighted partitions involving $s_k(π)$ with another parameter $z$, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving $s_k(π)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $k$th smallest part of a partition into distinct parts Gupta, Rajat Lebowitz-Lockard, Noah Vandehey, Joseph Number Theory Combinatorics Primary 11P81, 11P82, Secondary 11P84, 05A19 A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $π$ into distinct parts of any integer $n$, namely $s_k(π)$. Using $s_k(π)$, we generalize the above result for the $k$th smallest parts of partitions for any positive integer $k$ and show its connection with divisor functions for general $k$ and derive interesting special cases. We also study weighted partitions involving $s_k(π)$ with another parameter $z$, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving $s_k(π)$. |
| title | On the $k$th smallest part of a partition into distinct parts |
| topic | Number Theory Combinatorics Primary 11P81, 11P82, Secondary 11P84, 05A19 |
| url | https://arxiv.org/abs/2402.12549 |