Séminaire Lotharingien de Combinatoire, 80B.36 (2018), 12 pp.

Kevin Dilks, Jessica Striker and Corey Vorland

Increasing Labelings, Generalized Promotion and Rowmotion

Abstract. We generalize Bender-Knuth promotion on linear extensions to an analogous action on increasing labelings of any finite poset, in which the restrictions on the values of the labels satisfy a natural consistency condition. We give an equivariant bijection between such increasing labelings under this generalized promotion and order ideals in an associated poset under rowmotion. Additionally, we give a criterion for when certain kinds of toggle group actions on order ideals of a finite poset are conjugate to rowmotion. These results build upon work of O. Pechenik with the first two authors in the case of rectangular increasing tableaux and work of N. Williams with the second author relating promotion and rowmotion on ranked posets. We apply these results to posets embedded in the Cartesian product of ranked posets and increasing labelings with labels between 1 and q, in which case we obtain new instances of the resonance phenomenon.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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