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On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers
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On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers
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Paper
On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers
2026
Overview
Weighted Catalan numbers are a class of weighted sums over Dyck paths. Well-studied for their arithmetic properties and applications to enumerative combinatorics, these numbers were recently generalized to the setting of \\(k\\)-dimensional Catalan numbers for \\(k 2\\). In this paper, we introduce the \\(k\\)-dimensional semisymmetric weighted Catalan numbers (\\(k\\)-dimensional SSWCNs), an alternative \\(k\\)-dimensional generalization, along with their variant, the \\(k\\)-dimensional \\(u\\)-bounded semisymmetric weighted Catalan numbers (\\(k\\)-dimensional \\(u\\)-bounded SSWCNs). We define these two classes of numbers using the notion of semisymmetric height, a new statistic on points in \\(Z^k_ 0\\) motivated by geometric symmetries of \\(k\\)-dimensional analogs of Dyck paths and of the fundamental Weyl chamber of type \\(A_k-1\\). For our main results, we prove the eventual periodicity of \\(k\\)-dimensional SSWCNs and their \\(u\\)-bounded variants modulo a suitable integer \\(m\\), and we derive formulas for several classes of \\(k\\)-dimensional \\(u\\)-bounded SSWCNs. Additionally, using semisymmetric height, we derive novel analogs in the \\(k\\)-dimensional setting of the integer sequence counting Dyck paths by height and of the Narayana numbers. We conclude the paper with a future direction for generalizing weighted Catalan numbers to the \\(k\\)-dimensional setting.
Publisher
Cornell University Library, arXiv.org
Subject
