On a Subposet of the Tamari Lattice

TL;DR

This paper investigates the comb poset, a subposet of the Tamari lattice, revealing improved properties of certain functions within it and relating it to a known partial order on the symmetric group.

Contribution

It introduces the comb poset as a subposet of the Tamari lattice and demonstrates its advantageous properties for specific binary functions, connecting it to Edelman's partial order.

Findings

Rotation distance is well-behaved within the comb poset.

Meets and joins are well-behaved in the comb poset.

The common parse words function is well-behaved within the comb poset.

Abstract

We explore some of the properties of a subposet of the Tamari lattice introduced by Pallo, which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman.