The Delta Conjecture, valley version

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Let nn be a positive integer and let kk satisfy k≤nk\leq n. Let MPFn,k\mathcal{M P F}_{n,k} be the set of valley-marked parking functions of size nn with kk marked valleys and labels [n][n]. For such a parking function, let area⁡\operatorname{area}, dinv⁡\operatorname{dinv}, and ides⁡\operatorname{ides} denote its area, marked diagonal-inversion, and inverse-descent statistics, respectively. Let ene_n be the elementary symmetric function, let Δen−k−1′\Delta'_{e_{n-k-1}} denote the primed Delta operator indexed by en−k−1e_{n-k-1}, and let Fides⁡F_{\operatorname{ides}} be the fundamental quasisymmetric function indexed by the inverse-descent set. The Delta Conjecture, valley version. For all k≤nk\leq n,

Δen−k−1′en=∑MPF∈MPFn,ktarea⁡(MPF)qdinv⁡(MPF)Fides⁡(MPF).\Delta'_{e_{n-k-1}}e_n=\sum_{M P F\in\mathcal{M P F}_{n,k}}t^{\operatorname{area}(M P F)}q^{\operatorname{dinv}(M P F)}F_{\operatorname{ides}(M P F)}.

This is one of the principal combinatorial formulations of the Delta Conjecture; the case z=0z=0 recovers the Shuffle Theorem. Many special cases have been proved, but the general conjecture remains open.

References

Primary source

James Haglund and Emily Sergel, “Schedules and the Delta Conjecture”, arXiv:1908.04732 (2020).

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