Séminaire Lotharingien de Combinatoire, 89B.20 (2023), 12 pp.
Joseph Pappe, Stephan Pfannerer, Anne Schilling and
Mary Claire Simone
Promotion for Fans of Dyck Paths
Abstract.
We construct an injection from the set of r-fans of Dyck paths of length n into the set of chord diagrams on [n]
that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion matrices and in terms
of Fomin growth diagrams. Our analysis uses the fact that r-fans of Dyck paths can be viewed as highest weight
elements of weight zero in crystals of type Br, which in turn can be analyzed using virtual crystals. On the level of
Fomin growth diagrams, the virtualization process corresponds to the Roby-Krattenthaler blow up construction.
Our construction generalizes to vacillating tableaux as well.
We give a cyclic sieving phenomenon on r-fans of Dyck paths using the promotion action.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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