A proof of the symmetric theta trees conjecture when q = 0

Abstract

In [7], D'Adderio et al. conjecture a combinatorial formula for the expressions $\Xi e_{\alpha }{|}_{t=1}$, known as symmetric Theta trees Conjecture, in terms of tiered trees with an inversion statistic. In [18], Iraci and Romero 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 trees conjecture when $q=0$.