Meeting Covered Elements in $ν$-Tamari Lattices

Abstract

For each complete meet-semilattice M , we define an operator Pop M : M → M by Pop M (x) = ({y ∈ M : y x} ∪ {x}).When M is the right weak order on a symmetric group, Pop M is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of Pop Tam(ν) , where Tam(ν) is the ν-Tamari lattice. We determine the maximum size of a forward orbit of Pop Tam(ν) . When Tam(ν) is the n th m-Tamari lattice, this maximum forward orbit size is m + n − 1; in this case, we prove that the number of forward orbits of size m + n − 1 isMotivated by the recent investigation of the pop-stack-sorting map, we define a lattice path µ ∈ Tam(ν) to be t-Pop-sortable if Pop t Tam(ν) (µ) = ν. We enumerate 1-Pop-sortable lattice paths in Tam(ν) for arbitrary ν. We also give a recursive method to generate 2-Pop-sortable lattice paths in Tam(ν) for arbitrary ν; this allows us to enumerate 2-Pop-sortable lattice paths in a large variety of ν-Tamari lattices that includes the m-Tamari lattices.

Crystal Pop-Stack Sorting and Type A Crystal Lattices

“…where h is the Coxeter number of W . Results of similar flavors were obtained for semilattice pop-stack sorting operators on ν-Tamari lattices in [Def21a] and for (some) Coxeter pop-tsack torsing operators in [DW21].…”

“…where the cover relations and the meet are taken in the right weak order on W . The latter definition extends naturally to arbitrary complete meet-semilattices, yielding the notion of a semilattice pop-stack sorting operator that the first author explored in [Def21a]. The reason for using the name "pop-stack sorting" for these operators comes from the fact that Pop Sn coincides with the original pop-stack sorting map.…”

“…If the crystal B λ happens to be a lattice, then it comes equipped with two pop-stack sorting operators: the semilattice pop-stack sorting operator defined in [Def21a] and the crystal pop-stack sorting operator Pop ♦ . These two operators need not coincide; for example, they differ on the type A crystal B 2 (2,1) depicted in Figure 1.…”