The generalised valley version of the Delta conjecture

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Let m,n,k∈Nm,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 Δen−k−1′\Delta'_{e_{n-k-1}} be the Delta operators indexed by the complete and elementary symmetric functions. Generalised Delta conjecture, valley version.

ΔhmΔen−k−1′en=∑P∈LD(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.

References

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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