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

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 ${\mathcal{C}}_{n,A}^{\ast }$ in this paper. By establishing a labeled Dyck path model for the regions of ${\mathcal{C}}_{n,A}$ and ${\mathcal{C}}_{n,A}^{\ast }$, this paper explores several enumerative problems related to the number of regions of level $\mathrm{\ell }$, denoted as $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A})$ and $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A}^{\ast })$, which includes: (1) proving a Stirling convolution relation between $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A})$ and $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A}^{\ast })$, refining a result by Stanley and Postnikov; (2) showing that the sequences${\left(r_{\mathrm{\ell }}({\mathcal{C}}_{n,A})\right)}_{n\ge 0}$ and $(r_{\mathrm{\ell }}({\mathcal{C}}_{n,A}^{\ast }){)}_{n\ge 0}$ exhibit properties of binomial type in the sense of Rota; (3) establishing the transformational significance of $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A})$ and $r_{\mathrm{\ell }}({\mathcal{C}}_{n,A}^{\ast })$ 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 $\mathrm{\ell }$ 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.