Meeting covered elements in ν-Tamari lattices

Abstract

For each meet-semilattice M, we define an operator PopM:M→M byPopM(x)=⋀({y∈M:y⋖x}∪{x}). When M is the right weak order on a symmetric group, PopM 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 PopTam(ν), where Tam(ν) is the ν-Tamari lattice. We determine the maximum size of a forward orbit of PopTam(ν). When Tam(ν) is the nthm-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 is1n−1((m+1)(n−2)+m−1n−2). Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path μ∈Tam(ν) to be t-Pop-sortable if PopTam(ν)t(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.