Regions of Level $\ell$ of Catalan/Semiorder-Type Arrangements
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arXiv
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2024
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| author | Chen, Yanru Wang, Suijie Yang, Jinxing Zhao, Chengdong |
| author_facet | Chen, Yanru Wang, Suijie Yang, Jinxing Zhao, Chengdong |
| contents | In 1996, Stanley extended the classical Catalan arrangement and semiorder arrangement, which are called the Catalan-type arrangement $\mathcal{C}_{n,A}$ and the semiorder-type arrangement $\mathcal{C}_{n,A}^*$ in this paper. By establishing a labeled Dyck path model for the regions of $\mathcal{C}_{n,A}$ and $\mathcal{C}_{n,A}^*$, this paper explores several enumerative problems related to the number of regions of level $\ell$, denoted as $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$, which includes:
(1) proving a Stirling convolution relation between $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$, refining a result by Stanley and Postnikov;
(2) showing that the sequences$\left(r_\ell{(\mathcal{C}_{n,A})}\right)_{n\geq 0}$ and $(r_\ell {(\mathcal{C}_{n,A}^*)})_{n\geq 0}$ exhibit properties of binomial type in the sense of Rota;
(3) establishing the transformational significance of $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$ under Stanley's ESA framework: they can be viewed as transition matrices from binomial coefficients to their characteristic polynomials respectively.
Further, we present two applications of the theories and methods: first, we provide a hyperplane arrangement counting interpretation of the two-parameter generalization of Fuss--Catalan numbers, which is closely related to the number of regions of level $\ell$ in the $m$-Catalan arrangement. Second, using labeled Dyck paths to depict the number of regions in the $m$-Catalan arrangement, we algorithmically provide the inverse mapping of the Fu, Wang, and Zhu mapping. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regions of Level $\ell$ of Catalan/Semiorder-Type Arrangements Chen, Yanru Wang, Suijie Yang, Jinxing Zhao, Chengdong Combinatorics 05B35, 52C35 In 1996, Stanley extended the classical Catalan arrangement and semiorder arrangement, which are called the Catalan-type arrangement $\mathcal{C}_{n,A}$ and the semiorder-type arrangement $\mathcal{C}_{n,A}^*$ in this paper. By establishing a labeled Dyck path model for the regions of $\mathcal{C}_{n,A}$ and $\mathcal{C}_{n,A}^*$, this paper explores several enumerative problems related to the number of regions of level $\ell$, denoted as $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$, which includes: (1) proving a Stirling convolution relation between $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$, refining a result by Stanley and Postnikov; (2) showing that the sequences$\left(r_\ell{(\mathcal{C}_{n,A})}\right)_{n\geq 0}$ and $(r_\ell {(\mathcal{C}_{n,A}^*)})_{n\geq 0}$ exhibit properties of binomial type in the sense of Rota; (3) establishing the transformational significance of $r_{\ell}(\mathcal{C}_{n,A})$ and $r_{\ell}(\mathcal{C}_{n,A}^*)$ under Stanley's ESA framework: they can be viewed as transition matrices from binomial coefficients to their characteristic polynomials respectively. Further, we present two applications of the theories and methods: first, we provide a hyperplane arrangement counting interpretation of the two-parameter generalization of Fuss--Catalan numbers, which is closely related to the number of regions of level $\ell$ in the $m$-Catalan arrangement. Second, using labeled Dyck paths to depict the number of regions in the $m$-Catalan arrangement, we algorithmically provide the inverse mapping of the Fu, Wang, and Zhu mapping. |
| title | Regions of Level $\ell$ of Catalan/Semiorder-Type Arrangements |
| topic | Combinatorics 05B35, 52C35 |
| url | https://arxiv.org/abs/2410.10198 |