Analytic properties arising from the Baxter numbers
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916728722685952 |
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| author | Fang, Hanqian Zhang, Candice X. T. Zhao, James J. Y. |
| author_facet | Fang, Hanqian Zhang, Candice X. T. Zhao, James J. Y. |
| contents | Baxter numbers are known as the enumeration of Baxter permutations and numerous other discrete structures, playing a significant role across combinatorics, algebra, and analysis. In this paper, we focus on the analytic properties related to Baxter numbers. We prove that the descent polynomials of Baxter permutations have interlacing zeros, which is a property stronger than real-rootedness. Our approach is based on Dilks' framework of $(q,t)$-Hoggatt sums, which is a $q$-analog for Baxter permutations. Within this framework, we show that the family of $(1,t)$-Hoggatt sums satisfies the interlacing property using fundamental results on Hadamard products of polynomials. For Baxter numbers, we prove their asymptotic $r$-log-convexity via asymptotic expansions of $P$-recursive sequences. In particular, we confirm their $2$-log-convexity using symbolic computation techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic properties arising from the Baxter numbers Fang, Hanqian Zhang, Candice X. T. Zhao, James J. Y. Combinatorics 26C10, 11B83, 05A10 Baxter numbers are known as the enumeration of Baxter permutations and numerous other discrete structures, playing a significant role across combinatorics, algebra, and analysis. In this paper, we focus on the analytic properties related to Baxter numbers. We prove that the descent polynomials of Baxter permutations have interlacing zeros, which is a property stronger than real-rootedness. Our approach is based on Dilks' framework of $(q,t)$-Hoggatt sums, which is a $q$-analog for Baxter permutations. Within this framework, we show that the family of $(1,t)$-Hoggatt sums satisfies the interlacing property using fundamental results on Hadamard products of polynomials. For Baxter numbers, we prove their asymptotic $r$-log-convexity via asymptotic expansions of $P$-recursive sequences. In particular, we confirm their $2$-log-convexity using symbolic computation techniques. |
| title | Analytic properties arising from the Baxter numbers |
| topic | Combinatorics 26C10, 11B83, 05A10 |
| url | https://arxiv.org/abs/2505.05873 |