The pop-stack-sorting operator on Tamari lattices

Abstract

Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator PopM:M→M for each complete meet-semilattice M byPopM(x)=⋀({y∈M:y⋖x}∪{x}). This paper concerns the dynamics of PopTamn, where Tamn is the n-th Tamari lattice.We say an element x∈Tamn is t-Pop-sortable if PopMt(x) is the minimal element and we let ht(n) denote the number of t-Pop-sortable elements in Tamn. We find an explicit formula for the generating function ∑n≥1ht(n)zn and verify Defant's conjecture that it is rational. We furthermore prove that the size of the image of PopTamn is the Motzkin number Mn, settling a conjecture of Defant and Williams.