On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers

Abstract

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 $k \geq 2$. 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 $\mathbb{Z}^k_{\geq 0}$ 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.