The pop-stack-sorting operator on Tamari lattices

Counting smaller elements in the Tamari and m-Tamari lattices

An extension of Tamari lattices

For any finite path v on the square lattice consisting of north and east unit steps, we construct a poset Tam(v) that consists of all the paths lying weakly above v with the same endpoints as v. For particular choices of v, we recover the traditional Tamari lattice and the m-Tamari lattice. In particular this solves the problem of extending the m-Tamari lattice to any pair (a, b) of relatively ...

Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F , summarized in the fact that F is a group of fractions for a certain monoid F sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F sym. As an application, we show tha...

A stack and pop stack in series

We study sorting machines consisting of a stack and a pop stack in series, with or without a queue between them. While there are, a priori, four such machines, only two are essentially different: a pop stack followed directly by a stack, and a pop stack followed by a queue and then by a stack. In the former case, we obtain complete answers for the basis and enumeration of the sortable permutati...

Tamari lattices, forests and Thompson monoids

A connection relating Tamari lattices on symmetric groups regarded as lattices under the weak Bruhat order to the positive monoid P of Thompson group F is presented. Tamari congruence classes correspond to classes of equivalent elements in P. The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine the Tamari classes. In the monoid P thes...