Séminaire Lotharingien de Combinatoire, 80B.12 (2018), 12 pp.

Georgia Benkart, Laura Colmenarejo, Pamela E. Harris, Rosa Orellana, Greta Panova, Anne Schilling and Martha Yip

A minimaj-Preserving Crystal Structure on Ordered Multiset Partitions

Abstract. We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also yields a bijective proof of the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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