The modified valley version of the Delta square conjecture

Let n,kNn,k\in\mathbb{N} with k<nk<n, and let LSQ(n)k\mathsf{LSQ}'(n)^{\bullet k} denote the partially labelled square paths with kk decorated valleys. For such a path PP, let dinv(P)\mathsf{dinv}(P), area(P)\mathsf{area}(P), and xPx^P denote its diagonal-inversion statistic, area, and monomial weight. Modified Delta square conjecture, valley version.

Θekω(pnk)=PLSQ(n)kqdinv(P)tarea(P)xP.\Theta_{e_k} \nabla \omega(p_{n-k}) = \sum_{P \in \mathsf{LSQ}'(n)^{\bullet k}} q^{\mathsf{dinv}(P)} t^{\mathsf{area}(P)} x^P.

The source refers to this as the square analogue of the valley Delta conjecture and points to Iraci and Vanden Wyngaerd for its definitions and prior formulation. Its status is not resolved in the supplied source.

Sources & referencesView supporting material

Primary source

Alessandro Iraci and Anna Vanden Wyngaerd, “"Pushing" our way from the valley Delta to the generalised valley Delta”, arXiv:2101.02600 (2021).

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