A new generalization of the Narayana numbers inspired by linear operators on associative $d$-ary algebras

TL;DR

This paper introduces a broad generalization of Narayana numbers inspired by linear operators on associative d-ary algebras, unifying various combinatorial interpretations and extending classical results.

Contribution

It presents a new two-parameter array generalizing Narayana numbers, provides multiple combinatorial interpretations, and links these numbers to operator monomials in d-ary associative algebras.

Findings

Defines a generalized Narayana number $N_d(n,k)$ for $d \\geq 2$

Establishes combinatorial interpretations including paths and trees

Connects $N_d(n,k)$ to operator monomials in d-ary algebras

Abstract

We introduce and study a generalization of the Narayana numbers $N_d(n,k) = \frac{1}{n+1} \binom{n+1}{k+1} \binom{ n + (n-k)(d-2)+1}{k}$ for integers $d \geq 2$ and $n,k \geq 0$. This two-parameter array extends the classical Narayana numbers ($d=2$) and yields a $d$-ary analogue of the Catalan numbers $C_d(n) = \sum_{k=0}^n N_d(n,k)$. We give nine combinatorial interpretations of $N_d(n,k)$ that unify and generalize known combinatorial interpretations of the Narayana numbers and $C_3(n)$ in the literature. In particular, we show that $N_d(n,k)$ counts a natural class of operator monomials over a $d$-ary associative algebra, thereby extending a result of Bremner and Elgendy for the binary case. We also construct explicit bijections between these monomials and several families of classic combinatorial objects, including Schr\"{o}der paths, Dyck paths, rooted ordered trees, and $231$-avoiding permutations.