An identity relating Catalan numbers to tangent numbers with arithmetic applications
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918122486759424 |
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| author | Zhao, Tongyuan Lin, Zhicong Zang, Yongchun |
| author_facet | Zhao, Tongyuan Lin, Zhicong Zang, Yongchun |
| contents | We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ where $E_{2k+1}$ denote the tangent numbers. Interestingly, the latter identity can be applied to prove that $(n + 1)E_{2n+1}$ is divisible by $2^{2n}$ and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural $q$-analog of the latter identity with a combinatorial proof. This $q$-identity can be applied to prove Foata's divisibility property of the $q$-tangent numbers, which responds to a problem raised by Schützenberger. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An identity relating Catalan numbers to tangent numbers with arithmetic applications Zhao, Tongyuan Lin, Zhicong Zang, Yongchun Combinatorics Number Theory We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ where $E_{2k+1}$ denote the tangent numbers. Interestingly, the latter identity can be applied to prove that $(n + 1)E_{2n+1}$ is divisible by $2^{2n}$ and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural $q$-analog of the latter identity with a combinatorial proof. This $q$-identity can be applied to prove Foata's divisibility property of the $q$-tangent numbers, which responds to a problem raised by Schützenberger. |
| title | An identity relating Catalan numbers to tangent numbers with arithmetic applications |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2507.20965 |