The Haglund–Morse–Zabrocki nabla conjecture for compositions

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Let cc be a composition, let ∇\nabla be the nabla operator on symmetric functions, let Cc=Cc1⋯CckC_c=C_{c_1}\cdots C_{c_k} for c=[c1,…,ck]c=[c_1,\ldots,c_k], and let Ac\mathcal{A}_c be the set of parking functions with composition cc. The Haglund–Morse–Zabrocki nabla conjecture.

∇Cc1=∑PF∈Actarea⁡(PF)qdinv⁡(PF)Qides⁡(PF).\nabla C_c1=\sum_{PF\in\mathcal{A}_c}t^{\operatorname{area}(PF)}q^{\operatorname{dinv}(PF)}Q_{\operatorname{ides}(PF)}.

This conjecture connects the nabla action on modified Hall–Littlewood operator products with parking-function statistics and refines the shuffle-conjecture framework by composition. The source gives no resolution.

References

Primary source

Angela Hicks, “A Parking Function Bijection supporting the Haglund-Morse-Zabrocki Conjectures”, arXiv:1210.2705 (2012).

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