Séminaire Lotharingien de Combinatoire, B74d (2016), 17 pp.

Lukas Riegler and Christoph Neumann

Playing Jeu De Taquin on d-Complete Posets

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 Dm,n and show that uniform distribution is obtained if and only if Dm,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?

Received: May 7, 2014. Revised: August 6, 2016. Accepted: August 30, 2016.

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