Playing jeu de taquin on d-complete posets
Lukas Riegler, Christoph Neumann

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 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 and show that uniform distribution is obtained if and only if 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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Random Matrices and Applications
