Playing jeu de taquin on d-complete posets

TL;DR

This paper extends jeu de taquin to arbitrary posets, analyzes its uniformity on d-complete posets, and connects it to counting linear extensions and expected values in Young tableaux.

Contribution

It introduces a generalized jeu de taquin for posets, characterizes when it yields uniform distributions, and links this to the extended hook-length formula for d-complete posets.

Findings

Uniform distribution occurs if and only if the poset is d-complete.

Extended hook-length formula counts linear extensions of d-complete posets.

Connection established between jeu de taquin, linear extensions, and expected tableau entries.

Abstract

Using a modified version of jeu de taquin, Novelli, Pak and Stoyanovskii gave a bijective proof of the hook-length formula for counting standard Young tableaux of fixed shape. In this paper we consider a natural extension of jeu de taquin to arbitrary posets. Given a poset P, jeu de taquin defines a map from the set of bijective labelings of the poset elements with $\{1,2,...,|P|\}$ to the set of linear extensions of the poset. One question of particular interest is for which posets this map yields each linear extension equally often. We analyze the double-tailed diamond poset $D_{m,n}$ and show that uniform distribution is obtained if and only if $D_{m,n}$ is d-complete. Furthermore, we observe that the extended hook-length formula for counting linear extensions on d-complete posets provides a combinatorial answer to a seemingly unrelated question, namely: Given a uniformly random standard Young tableau of fixed shape, what is the expected value of the left-most entry in the second row?