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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 *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?

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

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