Cyclic Sieving of Increasing Tableaux and small Schröder Paths

Abstract

An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.

20 pages

References in corpus (2)

• Proofs and generalizations of a homomesy conjecture of Propp and Roby
• Graded cluster algebras

Cited by in corpus (20)

• Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics
• Resonance in orbits of plane partitions and increasing tableaux
• Proofs and generalizations of a homomesy conjecture of Propp and Roby
• Genomic Tableaux
• Cyclic Descents for General Skew Tableaux
• Rowmotion and Increasing Labeling Promotion
• A web basis of invariant polynomials from noncrossing partitions
• Orbits of Plane Partitions of Exceptional Lie Type
• The genomic Schur function is fundamental-positive
• Dynamics of plane partitions: Proof of the Cameron-Fon-Der-Flaass conjecture
• Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
• A skein action of the symmetric group on noncrossing partitions
• The cyclic sieving phenomenon on circular Dyck paths
• From Dyck paths to standard Young tableaux
• Enumeration on row-increasing tableaux of shape $2\times n$
• Tableau Stabilization and Lattice Paths
• On cyclic Schur-positive sets of permutation
• Set partitions, fermions, and skein relations
• Web invariants for flamingo Specht modules
• Chains of binary paths and shifted tableaux