Counting Dyck paths by area and rank

The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting $(q,t)$-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: one refines an identity of Carlitz and Riordan; the other refines the notion of $\gamma$-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane's correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the $n$-cycle $(12...n)$ in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump (arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first draf

Refined Catalan and Narayana cyclic sieving

We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.Comment: Updated version, minor change

Parking Functions And Generalized Catalan Numbers

Since their introduction by Konheim and Weiss, parking functions have evolved into objects of surprising combinatorial complexity for their simple definitions. First, we introduce these structures, give a brief history of their development and give a few basic theorems about their structure. Then we examine the internal structures of parking functions, focusing on the distribution of descents and inversions in parking functions. We develop a generalization to the Catalan numbers in order to count subsets of the parking functions. Later, we introduce a generalization to parking functions in the form of k-blocked parking functions, and examine their internal structure. Finally, we expand on the extension to the Catalan numbers, exhibiting examples to explore its internal structure. These results continue the exploration of the deep structures of parking functions and their relationship to other combinatorial objects

Enumeration of non-crossing partitions according to subwords with repeated letters

An avoidance pattern where the letters within an occurrence of which are required to be adjacent is referred to as a subword. In this paper, we enumerate members of the set NC_n of non-crossing partitions of length n according to the number of occurrences of several infinite families of subword patterns each containing repeated letters. As a consequence of our results, we obtain explicit generating function formulas counting the members of NC_n for n >= 0 according to all subword patterns of length three containing a repeated letter. Further, simple expressions are deduced for the total number of occurrences over all members of NC_n for the various families of patterns. Finally, combinatorial proofs can be given explaining three infinite families of subword equivalences over NC_n, which generalize the following equivalences: 211 = 221, 1211 = 1121 and 112 = 122

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds