A note on Tamari intervals

TL;DR

This paper investigates symmetry properties of a polynomial enumerating intervals in Tamari lattices, revealing new symmetries and connections to binary planar trees, advancing combinatorial understanding of these structures.

Contribution

It establishes a ternary symmetry for the polynomial associated with Tamari lattices and conjectures a broader global symmetry, linking interval enumeration to tree canopy statistics.

Findings

Proves a ternary symmetry in the polynomial for Tamari lattices.

Describes the set of synchronized intervals via Newton polytope facets.

Connects interval enumeration to canopy statistics of binary planar trees.

Abstract

To every partial order P, one associates a polynomial $\mathbb{D}_P$ in 4 variables that enumerates the intervals of P according to 4 parameters. Some symmetry properties of this polynomial are obtained for a specific family of posets, the Tamari lattices. A ternary symmetry is proved for the polynomial in 3 variables obtained by setting one variable to 1. Another global symmetry is conjectured. The set of synchronized intervals is described using a facet of the Newton polytope. A relation to the statistics of the canopy of binary planar trees is described.