The image of the pop operator on various lattices

Abstract

Extending the classical pop-stack sorting map on the lattice given by the right weak order on Sn, Defant defined, for any lattice M, a map PopM:M→M that sends an element x∈M to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study PopM(M) when M is the weak order of type Bn, the Tamari lattice of type Bn, the lattice of order ideals of the root poset of type An, and the lattice of order ideals of the root poset of type Bn. In particular, we settle four conjectures proposed by Defant and Williams on the generating functionPop(M;q)=∑b∈PopM(M)q|UM(b)|, where UM(b) is the set of elements of M that cover b.