Exceptional and modern intervals of the Tamari lattice

TL;DR

This paper characterizes special families of intervals in the Tamari lattice using interval-posets, linking them to noncrossing trees, noncrossing partitions, and ternary trees, and introduces the concepts of exceptional, modern, and infinitely modern intervals.

Contribution

It introduces and characterizes exceptional, modern, and infinitely modern interval-posets, connecting Tamari lattice intervals to noncrossing structures and ternary trees.

Findings

Exceptional intervals correspond to noncrossing trees and noncrossing partitions.

Modern intervals are in bijection with new intervals of the Tamari lattice.

Infinitely modern intervals match the count of ternary trees with n inner vertices.

Abstract

In this article we use the theory of interval-posets recently introduced by Ch{\^a}tel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider what we call exceptional interval-posets and show that they correspond to the intervals which are obtained as images of noncrossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattice induced by intervals in the poset of noncrossing partitions. In the second part we introduce the notion of modern and infinitely modern interval-posets. We show that the modern intervals are in bijection with the new intervals of the Tamari lattice in the sense of Chapoton. We deduce an intrinsic characterization of the new intervals in the Tamari lattice. Finally, we consider the family of what we call infinitely modern intervals and we we prove that there are as many infinitely modern interval-posets of size n as there are ternary trees with n inner vertices.