Signature Catalan combinatorics

Coxeter-Catalan Combinatorics

In this talk we will introduce the new and active topic “CoxeterCatalan combinatorics”. We will survey three trends, namely the nonnesting partitions, noncrossing partitions and cluster complexes, leading to this topic. Some new results and unsolved problems will be given.

Rational Catalan Combinatorics: The Associahedron

Each positive rational number x > 0 can be written uniquely as x = a/(b − a) for coprime positive integers 0 < a < b. We will identify x with the pair (a, b). In this extended abstract we use rational Dyck paths to define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a, b) called the rational associahedron. It is a pure simplicial complex of dimension a− 2, and its maximal ...

Catalan structures and Catalan pairs

A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms. These objects are enumerated by the well-known Catalan numbers, and have been introduced in [DFPR] with the aim of giving a common language to most of the structures counted by Catalan numbers. Here, we give a simple method to pass from the recursive definition of a generic Catalan structure to the recursive definitio...

Coincidences of Catalan and Q-catalan Numbers

Let Cn and Cn(q) be the nth Catalan number and the nth q-Catalan number, respectively. In this paper, we show that the Diophantine equation Cn = Cm(q) has only finitely many integer solutions (m,n, q) with m > 1, n > 1, q > 1. Moreover, they are all effectively computable. – To Professor Carl Pomerance on his 65th birthday

Combinatorics Enumerative Combinatorics 05