Cuntz-Krieger algebras and a generalization of Catalan numbers
Abstract
We first observe that the relations of the canonical generating isometries of the Cuntz algebra
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
O
A
G
for the transition matrix
A
G
of
G
. The generalized Catalan numbers
c
G
n
,n=0,1,2,...
enumerate the number of Dyck paths and oriented rooted trees for the graph
G
. Its generating functions will be studied.