Regions of Level $\ell$ of Catalan/Semiorder-Type Arrangements

Abstract: 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.