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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