A Proof of the Symmetric Theta Conjecture when q = 0

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 $q=0$.