Promotion of Kreweras words

Abstract

Kreweras words are words consisting of nmathrm {A}’s, nmathrm {B}’s, and nmathrm {C}’s in which every prefix has at least as many mathrm {A}’s as mathrm {B}’s and at least as many mathrm {A}’s as mathrm {C}’s. Equivalently, a Kreweras word is a linear extension of the poset mathsf{V}times [n]. Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the mathrm {B}’s and mathrm {C}’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s mathfrak {sl}_3-webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.