The generalised valley version of the Delta conjecture

Let m,n,kNm,n,k\in\mathbb{N} with k<nk<n, and let LD(m,n)k\mathsf{LD}(m,n)^{\bullet k} denote the partially labelled rectangular Dyck 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, respectively. Let Δhm\Delta_{h_m} and Δenk1\Delta'_{e_{n-k-1}} be the Delta operators indexed by the complete and elementary symmetric functions. Generalised Delta conjecture, valley version.

ΔhmΔenk1en=PLD(m,n)kqdinv(P)tarea(P)xP.\Delta_{h_m}\Delta'_{e_{n-k-1}} e_n = \sum_{P\in \mathsf{LD}(m,n)^{\bullet k}}q^{\mathsf{dinv}(P)}t^{\mathsf{area}(P)}x^{P}.

The source notes that this formula contains the preceding valley Delta conjecture as the special case m=0m=0 and attributes it to Qiu and Wilson. 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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