The image of the pop operator on various lattices

Extending the classical pop-stack sorting map on the lattice given by the right weak order on S-n, Defant defined, for any lattice M, a map Pop(M) : M -> M that sends an element x is an element of 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 Pop(M) (M) when M is the weak order of type B-n, the Tamari lattice of type B-n, the lattice of order ideals of the root poset of type A(n), and the lattice of order ideals of the root poset of type B-n. In particular, we settle four conjectures proposed by Defant and Williams on the generating function Pop(M; q) = Sigma (b is an element of PopM (M)) q(|UM(b)|), where U-M (b) is the set of elements of M that cover b. \(c) 2023 Elsevier Inc. All rights reserved.