Promotion of Kreweras words

TL;DR

This paper explores the combinatorial and algebraic properties of Kreweras words, revealing their behavior under promotion, connecting them to web diagrams and answering a longstanding question about posets with 'good' promotion behavior.

Contribution

It demonstrates that promotion on Kreweras words swaps B's and C's after 3n applications and links Kreweras words to web diagrams and trip permutations, providing new insights into their structure.

Findings

Promotion swaps B's and C's after 3n applications.

Kreweras words are described via Kuperberg's webs and Postnikov's trip permutations.

First answer to Stanley's 2009 question on posets with 'good' promotion behavior.

Abstract

Kreweras words are words consisting of n A's, n B's, and n C's in which every prefix has at least as many A's as B's and at least as many A's as C's. Equivalently, a Kreweras word is a linear extension of the poset ${\sf V}\times [n]$. 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 Sch\"{u}tzenberger'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 B's and 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, Sch\"{u}tzenberger's promotion corresponds to rotation of the web.