The Delta Conjecture for the modified Macdonald delta operator

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Q}(q,t)$, let $e_k$ and $e_n$ be elementary symmetric functions, and let $\Delta'_f$ be the Macdonald eigenoperator defined by

$$\Delta'_f(\tilde{H}_{\mu})=f[B_{\mu}(q,t)-1]\tilde{H}_{\mu}.$$

For a labelled Dyck path $P\in\mathcal{LD}_n$, let $\operatorname{dinv}(P)$, $\operatorname{area}(P)$, $a_i(P)$, $d_i(P)$, $\operatorname{Val}(P)$, and $x^P$ denote the statistics and monomial associated with $P$ as in the source. The operator $\{z^r\}$ extracts the coefficient of $z^r$.

The Delta Conjecture. For any integers $n>k\geq 0$,

$$\Delta'_{e_k}e_n=\{z^{n-k-1}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i:a_i(P)>a_{i-1}(P)}\left(1+z/t^{a_i(P)}\right)x^P\right]$$

and

$$\Delta'_{e_k}e_n=\{z^{n-k-1}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i\in\operatorname{Val}(P)}\left(1+z/t^{d_i(P)+1}\right)x^P\right].$$

This is an open generalization of the Shuffle Conjecture, giving a combinatorial formula for a Macdonald delta-operator specialization. The paper proves a specialization of the conjecture using statistics on ordered set partitions, while the full statement remains open in the source.