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Cuntz-Krieger algebras and a generalization of Catalan numbers

Kengo Matsumoto
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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.

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Kengo Matsumoto. 2006-08-18. Cuntz-Krieger algebras and a generalization of Catalan numbers. https://arxiv.org/abs/math/0607517

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