The Pop-stack-sorting Operator on Tamari Lattices

Abstract

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