Bijective proof of a conjecture on unit interval posets

TL;DR

This paper provides a bijective proof of a conjecture relating two representations of unit interval posets via the zeta map, using a reformulation with left-aligned colored trees, advancing combinatorial understanding.

Contribution

It introduces a bijective proof of a conjecture on unit interval posets, employing a novel reformulation of the zeta map with colored trees.

Findings

Established a bijective correspondence confirming the conjecture

Reformulated the zeta map using left-aligned colored trees

Enhanced combinatorial tools for studying unit interval posets

Abstract

In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by G\'elinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.