Year: 2026
Type: preprint
Abstract: In this paper, we study the regions of asymmetric extensions of the Catalan and Shi arrangements. We first determine the characteristic polynomials using the finite field method, thereby obtaining explicit formulas for the number of regions. We then introduce two new classes of combinatorial objects: $\mathbf{c}$-labelled Dyck paths and $\mathbf{c}$-building functions, which generalize classical Dyck paths and parking functions, respectively. We prove that the regions of the asymmetric $\mathbf{c}$-Catalan arrangement are in natural bijection with $\mathbf{c}$-labelled Dyck paths, while those of the asymmetric $\mathbf{c}$-Shi arrangement are in natural bijection with $\mathbf{c}$-building functions. This resolves a problem proposed by Theo Douvropoulos during the 2022 Oberwolfach Workshop on Enumerative Combinatorics. In particular, our results extend Stanley's celebrated correspondence between $k$-parking functions and the regions of the $k$-Shi arrangement by introducing new combinatorial models and developing entirely different proof techniques.
Source: arXiv (Cornell University)
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202186476
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