Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a conjecture of Hopkins that the rectangle and trapezoid posets have the same rowmotion orbit structures. Our main tools in proving this are $K$-jeu-de-taquin and (weak) $K$-Knuth equivalence of increasing tableaux. We define almost minimal tableaux as a family of tableaux naturally arising from order ideals and show for any $\lambda$, the almost minimal tableaux of shape $\lambda$ are in different (weak) $K$-Knuth equivalence classes. We also discuss and make some progress on related conjectures of Hopkins on down-degree homomesy.
@article{10_37236_9769,
author = {Quang Vu Dao and Julian Wellman and Calvin Yost-Wolff and Sylvester W. Zhang},
title = {Rowmotion orbits of trapezoid posets},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/9769},
zbl = {1491.05204},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9769/}
}
TY - JOUR
AU - Quang Vu Dao
AU - Julian Wellman
AU - Calvin Yost-Wolff
AU - Sylvester W. Zhang
TI - Rowmotion orbits of trapezoid posets
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/9769/
DO - 10.37236/9769
ID - 10_37236_9769
ER -
%0 Journal Article
%A Quang Vu Dao
%A Julian Wellman
%A Calvin Yost-Wolff
%A Sylvester W. Zhang
%T Rowmotion orbits of trapezoid posets
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/9769/
%R 10.37236/9769
%F 10_37236_9769
Quang Vu Dao; Julian Wellman; Calvin Yost-Wolff; Sylvester W. Zhang. Rowmotion orbits of trapezoid posets. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/9769
