The compositional shuffle conjecture

Let $X=\{x_1,x_2,\ldots\}$ be an infinite set of variables, and write $x_w=x_{w_1}\cdots x_{w_n}$. Let $\alpha=(\alpha_1,\ldots,\alpha_k)$ be a composition of $n$. For a Dyck path $\pi$ of semilength $n$, let $\operatorname{touch}(\pi)=\alpha$ mean that the successive gaps between its contacts with the main diagonal have lengths $\alpha_1,\ldots,\alpha_k$. Let $\mathcal{WP}_\pi$ be the associated word parking functions, and let $\operatorname{area}(\pi)$ and $\operatorname{dinv}(\pi,w)$ be their statistics. The operator $C_\alpha$ is the compositional creation operator, and $C_\alpha(1)$ denotes its value at $1$.

The compositional shuffle conjecture. For any composition $\alpha$,

$$(-1)^n\nabla C_\alpha(1)=\sum_{\operatorname{touch}(\pi)=\alpha}t^{\operatorname{area}(\pi)}\sum_{w\in\mathcal{WP}_\pi}q^{\operatorname{dinv}(\pi,w)}x_w.$$

This refines the original shuffle conjecture by partitioning the Dyck-path sum according to the touch composition. It was proved in the Carlsson–Mellit work underlying these lecture notes, so the conjecture is solved.