The Delta Conjecture, valley version
Let be a positive integer and let satisfy . Let be the set of valley-marked parking functions of size with marked valleys and labels . For such a parking function, let , , and denote its area, marked diagonal-inversion, and inverse-descent statistics, respectively. Let be the elementary symmetric function, let denote the primed Delta operator indexed by , and let be the fundamental quasisymmetric function indexed by the inverse-descent set. The Delta Conjecture, valley version. For all ,
This is one of the principal combinatorial formulations of the Delta Conjecture; the case 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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