Canopy of binary trees, Catalan tableaux and the asymmetric exclusion process

Abstract

The purpose of this paper is twofold. First we answer to a question asked by Steingrimsson and Williams about certain permutation tableaux: we construct a bijection between binary trees and the so-called Catalan tableaux. These tableaux are certain Ferrers (or Young) diagrams filled with some 0's and 1's, satisfying a certain hook condition, and are enumerated by the Catalan numbers. They form a subclass of the permutation tableaux, enumerated by n!, introduced by Postnikov in his study of totally non negative Grassmannians and networks. Secondly we relate this new Catalan bijection with the totally asymmetric exclusion process (TASEP), a very rich and well studied 1D gas model in statistical mechanics of nonequilibrium systems. We continue some combinatorial understanding of that model, in the spirit of works by Shapiro, Zeilberger and more recently by Brak, Essam, Rechnitzer, Corteel, Williams, Duchi and Schaeffer. Emphasis is made on the non-classical notion of canopy of a binary tree, analog of the classical up-down sequence of a permutation.