Rowmotion on $m$-Tamari and BiCambrian Lattices

TL;DR

This paper proves conjectures about the order and orbit structure of rowmotion on certain Tamari and biCambrian lattices, revealing cyclic sieving phenomena and homomesy properties in these combinatorial structures.

Contribution

It constructs an equivariant bijection to prove the order conjecture for rowmotion on rational Tamari lattices and characterizes the orbit structure on biCambrian lattices for specific Coxeter elements.

Findings

Rowmotion has order a+b-1 on certain Tamari lattices when b ≡ 1 mod a.

The orbit structure of rowmotion exhibits cyclic sieving in these cases.

The down-degree statistic is homomesic under rowmotion.

Abstract

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 $b\equiv 1\pmod a$; 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 $W$.