$P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors

We define P -strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P -strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.

[1]  ANATOL N KIRILLOV,et al.  GROUPS GENERATED BY INVOLUTIONS GELFAND TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX , 2011 .

[2]  P. Campbell,et al.  Hook-content Formulae for Symplectic and Orthogonal Tableaux , 2007, Canadian Mathematical Bulletin.

[3]  Brendon Rhoades,et al.  Cyclic sieving, promotion, and representation theory , 2010, J. Comb. Theory, Ser. A.

[4]  Tom Roby,et al.  Dynamical algebraic combinatorics and the homomesy phenomenon , 2016 .

[5]  Sen-Peng Eu,et al.  The cyclic sieving phenomenon for faces of generalized cluster complexes , 2008, Adv. Appl. Math..

[6]  Victor Reiner,et al.  The cyclic sieving phenomenon , 2004, J. Comb. Theory A.

[7]  Peter J. Cameron,et al.  Orbits of antichains revisited , 1995, Eur. J. Comb..

[8]  Tom Roby,et al.  Homomesy in Products of Two Chains , 2013, Electron. J. Comb..

[9]  S. Hopkins.  Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion , 2019, Exp. Math..

[10]  S. Hopkins.  Cyclic Sieving for Plane Partitions and Symmetry , 2019, 1907.09337.

[11]  Nathan Williams,et al.  Promotion and rowmotion , 2011, Eur. J. Comb..

[12]  Hugh Thomas,et al.  A uniform bijection between nonnesting and noncrossing partitions , 2011, 1101.1277.

[13]  Emden R. Gansner.  On the equality of two plane partition correspondences , 1980, Discret. Math..

[14]  Paths to Understanding Birational Rowmotion on Products of Two Chains , 2018, Algebraic Combinatorics.

[15]  Michelle L. Wachs,et al.  Flagged Schur Functions, Schubert Polynomials, and Symmetrizing Operators , 1985, J. Comb. Theory, Ser. A.

[16]  Edward A. Bender,et al.  Enumeration of Plane Partitions , 1972, J. Comb. Theory A.

[17]  Combinatorial, piecewise-linear, and birational homomesy for products of two chains , 2013, 1310.5294.

[18]  Gabriel Frieden,et al.  Affine type A geometric crystal on the Grassmannian , 2017, J. Comb. Theory A.

[19]  Alexander Schrijver,et al.  On the period of an operator, defined on antichains , 1974 .

[20]  Victor Reiner,et al.  What is... cyclic sieving , 2014 .

[21]  Cesar Ceballos,et al.  Subword complexes, cluster complexes, and generalized multi-associahedra , 2011, 1108.1776.

[22]  Jessica Striker.  Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance , 2017 .

[23]  Richard P. Stanley,et al.  Promotion and Evacuation , 2008, Electron. J. Comb..

[24]  Jessica Striker,et al.  Resonance in orbits of plane partitions and increasing tableaux , 2015, J. Comb. Theory, Ser. A.

[25]  J. Propp,et al.  Piecewise-linear and birational toggling , 2014, 1404.3455.

[26]  R. Stanley.  Ordered Structures And Partitions , 1972 .

[27]  Jessica Striker,et al.  Rowmotion and increasing labeling promotion , 2019, J. Comb. Theory, Ser. A.

[28]  Pierre Duchet.  Sur les hypergraphes invariants , 1974, Discret. Math..

[29]  Christian Krattenthaler,et al.  Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters , 1997, J. Comb. Theory A.

[30]  Ira M. Gessel,et al.  Determinants, Paths, and Plane Partitions , 1989 .

[31]  Shahrzad Haddadan.  Some Instances of Homomesy Among Ideals of Posets , 2021, Electron. J. Comb..

[32]  Luis Serrano,et al.  Maximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials , 2010, Electron. J. Comb..

[33]  Tom Roby,et al.  Iterative Properties of Birational Rowmotion II: Rectangles and Triangles , 2015, Electron. J. Comb..

[34]  Dan Saracino,et al.  Proofs and generalizations of a homomesy conjecture of Propp and Roby , 2016, Discret. Math..

[35]  Marcel Paul Schützenberger,et al.  Promotion des morphismes d'ensembles ordonnes , 1972, Discret. Math..