Linear Intervals in the Tamari and the Dyck Lattices and in the alt-Tamari Posets

TL;DR

This paper counts linear intervals in Tamari and Dyck lattices, finds they are numerically identical, and introduces a new family of posets called alt-Tamari posets, which generalize these lattices.

Contribution

It introduces alt-Tamari posets, a new family of posets on Dyck paths, and proves they share the same number of linear intervals as Tamari and Dyck lattices.

Findings

Number of linear intervals in Tamari and Dyck lattices are equal.

Alt-Tamari posets generalize Tamari and Dyck lattices.

All alt-Tamari posets have identical counts of linear intervals by height.

Abstract

We count the number of linear intervals in the Tamari and the Dyck lattices according to their height, using generating series and Lagrange inversion. Surprisingly, these numbers are the same in both lattices. We define a new family of posets on Dyck paths, which we call alt-Tamari posets. Each alt-Tamari poset depends on the choice of an increment function delta in {0,1}^n. We recover the Tamari and the Dyck lattices as extreme cases with delta = 1 and delta = 0, respectively. We prove that all the alt-Tamari posets have the same number of linear intervals of any given height.