Counting smaller trees in the Tamari order

TL;DR

This paper introduces interval-posets to encode Tamari lattice intervals, providing a combinatorial interpretation of a bilinear form and establishing a recursive polynomial that counts trees smaller than a given tree in the Tamari order.

Contribution

It presents a novel combinatorial object, interval-posets, and connects them to Tamari intervals, offering new insights into the structure and enumeration of Tamari lattice elements.

Findings

Interval-posets encode Tamari lattice intervals.

The polynomial from the bilinear form counts trees smaller than a given tree.

The functional equation for Tamari intervals is recovered and proved.

Abstract

We introduce new objects, the interval-posets, that encode intervals of the Tamari lattice. We then find a combinatorial interpretation of the bilinear form that appears in the functional equation of Tamari intervals described by Chapoton. Thus, we retrieve this functional equation and prove that the polynomial recursively computed from the bilinear form on each tree $T$ counts the number of trees smaller than $T$ in the Tamari order.