Rowmotion on \(m\)-Tamari and biCambrian lattices

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 \(a≡ 1 b\); 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 \(W\).