Loehr–Warrington Square Paths Conjecture

Let ${\cal P}ref_n$ be the set of all preference functions on $n$ cars. For a preference function $Pr$, let $\operatorname{area}(Pr)$ be its area, $\operatorname{dinv}(Pr)$ its diagonal-inversion number, and $\operatorname{ides}(\sigma(Pr))$ the inverse descent set of its word. Let $F_{\operatorname{ides}(\sigma(Pr))}$ denote the corresponding fundamental quasisymmetric function, and let $\nabla$ and $p_n$ denote the nabla operator and the $n$th power-sum symmetric function.

Loehr–Warrington's Square Paths Conjecture.

$$(-1)^{n-1} \nabla p_n = \sum_{Pr \in {\cal P}ref_n} t^{\operatorname{area}(Pr)} q^{\operatorname{dinv}(Pr)} F_{\operatorname{ides}(\sigma(Pr))}.$$

The paper states that its main result proves this conjecture, using enumerations of preference functions and their relation to parking functions. Thus the conjecture is solved by the paper.