Canonical Characters on Quasi-Symmetric Functions and Bivariate Catalan Numbers

Robin Sulzgruber (a0702809)

We introduce the q, t-Catalan numbers as the bivariate generating polynomials of two statistics on Dyck paths. We discuss some of their properties from a combinatorial point of view, e.g. a description by means of different statistics, a recursion, specialisations, and a possible generalisation to parking functions. We then use the fact that the q, tCatalan numbers are symmetric in q and t to c...

Cuntz-krieger Algebras and a Generalization of Catalan Numbers

We first observe that the relations of the canonical generating isometries of the Cuntz algebra ON 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 AG for the transition matrix A of G. The generalized Catalan numbers cGn , n = 0, 1, 2, . . . enumer...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern p we provide a bivariate generating function where the coefficient of xy in its serie...

Square-bounded partitions and Catalan numbers

For each integer k ≥ 1, we define an algorithm which associates to a partition whose maximal value is at most k a certain subset of all partitions. In the case when we begin with a partition λ which is square-bounded, i.e. λ= (λ1 ≥ · · · ≥ λk) with λ1 = k and λk = 1, applying the algorithm times gives rise to a set whose cardinality is either the Catalan number c −k+1 (the self dual case) or tw...