2022/08/22 by Colin Defant, Defant, Colin, James C. Lin +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2208.10464
Thomas and Williams conjectured that rowmotion acting on the rational (a,b)-Tamari lattice has order a+b-1. We construct an equivariant bijection that proves this conjecture when b≡ 1\pmod a; in fact, we determine the entire orbit structure of rowmotion in this case, showing that it exhibits the cyclic sieving phenomenon. We additionally show that the down-degree statistic is homomesic for this action. In a different vein, we consider the action of rowmotion on Barnard and Reading's biCambrian lattices. Settling a different conjecture of Thomas and Williams, we prove that if c is a bipartite Coxeter element of a coincidental-type Coxeter group W, then the orbit structure of rowmotion on the c-biCambrian lattice is the same as the orbit structure of rowmotion on the lattice of order ideals of the doubled root poset of type W.