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.
Highlights
In order to study evacuation of Kreweras words, we will employ another formulation of promotion and evacuation of linear extensions in terms of growth diagrams
A question we will answer in the following subsection is: which webs W are equal to Ww for some Kreweras word w? As we will see, the restriction coming from Proposition 5.3 is the only restriction
To previous work on webs and promotion. It seems that viewing a web as a plabic graph in order to extract a trip permutation is a new idea, Lam [18] and Fraser, Lam and Le [7] discuss some relationships between webs and plabic graphs
Summary
The famous ballot problem, whose history stretches back to the 19th century, asks in how many ways we can order the ballots of an election between two candidates Alice.
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