Modified Delta square conjecture, valley version

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Let n,kn,k be nonnegative integers, let pnp_n be the power-sum symmetric function, ω\omega the standard involution, and Θek\Theta_{e_k} and ∇\nabla the Theta and nabla operators. Let LSQ′(n)∙k\mathsf{LSQ}'(n)^{\bullet k} be the modified set of labelled valley-decorated square paths defined in the paper; for π\pi in this set, write dinv(π)\mathsf{dinv}(\pi), area(π)\mathsf{area}(\pi), and xπx^\pi for its associated statistics and monomial weight.

Modified Delta square conjecture, valley version.

Θek∇ω(pn−k)=∑π∈LSQ′(n)∙kqdinv(π)tarea(π)xπ.\Theta_{e_k} \nabla \omega(p_{n-k}) = \sum_{\pi \in \mathsf{LSQ}'(n)^{\bullet k}} q^{\mathsf{dinv}(\pi)} t^{\mathsf{area}(\pi)} x^\pi.

The paper describes this as a new formulation without a multiplicative correcting factor and says that it extends naturally to the generalised case. It remains open.

References

Primary source

Alessandro Iraci and Anna Vanden Wyngaerd, “A valley version of the Delta square conjecture”, arXiv:2003.12048 (2020).

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