Cuntz-Krieger algebras and a generalization of Catalan numbers

TL;DR

This paper explores the connection between Cuntz-Krieger algebras and generalized Catalan numbers, linking algebraic relations to combinatorial structures like Dyck paths and rooted trees.

Contribution

It introduces a generalization of Catalan numbers using Cuntz-Krieger algebras for directed graphs, extending combinatorial enumeration.

Findings

Generalized Catalan numbers count Dyck paths and rooted trees for a graph

Generating functions of these numbers are analyzed

Relations of Cuntz algebra generators relate to combinatorial structures

Abstract

We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$. Its generating functions will be studied.