A bijection proof of Andrews-Merca integer partition theorem
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911907544301568 |
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| author | Liu, Ji-Cai |
| author_facet | Liu, Ji-Cai |
| contents | Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on generating functions. Motivated by Andrews and Merca's results, we define six statistics related to the partitions of $n$ and show that the two triples of the six statistics are equidistributed. From this equidistributed result, we derive modulo $m$ extensions of Andrews and Merca's results for all integers $m\ge 2$. The proof of the main result is based on a general bijection on the set of partitions of $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A bijection proof of Andrews-Merca integer partition theorem Liu, Ji-Cai Combinatorics Number Theory 05A17, 05A19 Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on generating functions. Motivated by Andrews and Merca's results, we define six statistics related to the partitions of $n$ and show that the two triples of the six statistics are equidistributed. From this equidistributed result, we derive modulo $m$ extensions of Andrews and Merca's results for all integers $m\ge 2$. The proof of the main result is based on a general bijection on the set of partitions of $n$. |
| title | A bijection proof of Andrews-Merca integer partition theorem |
| topic | Combinatorics Number Theory 05A17, 05A19 |
| url | https://arxiv.org/abs/2405.00063 |