A Proof of the Symmetric Theta Conjecture when q = 0

TL;DR

This paper proves the Symmetric Theta Trees Conjecture for q=0 by establishing a bijection between tiered trees and doubly labelled Dyck paths with zero statistics, confirming a combinatorial formula for symmetric functions.

Contribution

It provides the first proof of the conjecture at q=0 by explicitly connecting two combinatorial models through a bijection.

Findings

Confirmed the conjecture for q=0 case.

Established a bijection between tiered trees and Dyck paths.

Validated the combinatorial formula for symmetric functions.

Abstract

In 10.1093/imrn/rnac258, the authors conjecture a combinatorial formula for the expressions $\Xi e_\alpha \rvert_{t=1}$, known as Symmetric Theta Trees Conjecture, in terms of tiered trees with an inversion statistic. In 10.1017/fms.2024.14, the authors prove a combinatorial formula for the same symmetric function, in terms of doubly labelled Dyck paths with the area statistic. In this paper, we give an explicit bijection between the subsets of the two families of objects when the relevant statistic is equal to $0$, thus proving the Symmetric Theta Tree Conjecture when $q=0$.