Geodesic
A minimaj-Preserving Crystal Structure on Ordered Multiset Partitions
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a11,
     author = {Georgia Benkart and Laura Colmenarejo and Pamela E. Harris and Rosa Orellana and Greta Panova and Anne Schilling and Martha Yip},
     title = {A {minimaj-Preserving} {Crystal} {Structure} on {Ordered} {Multiset} {Partitions}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a11/}
}
TY  - JOUR
AU  - Georgia Benkart
AU  - Laura Colmenarejo
AU  - Pamela E. Harris
AU  - Rosa Orellana
AU  - Greta Panova
AU  - Anne Schilling
AU  - Martha Yip
TI  - A minimaj-Preserving Crystal Structure on Ordered Multiset Partitions
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a11/
ID  - SLC_2018_80B_a11
ER  - 
%0 Journal Article
%A Georgia Benkart
%A Laura Colmenarejo
%A Pamela E. Harris
%A Rosa Orellana
%A Greta Panova
%A Anne Schilling
%A Martha Yip
%T A minimaj-Preserving Crystal Structure on Ordered Multiset Partitions
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a11/
%F SLC_2018_80B_a11
Georgia Benkart; Laura Colmenarejo; Pamela E. Harris; Rosa Orellana; Greta Panova; Anne Schilling; Martha Yip. A minimaj-Preserving Crystal Structure on Ordered Multiset Partitions. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a11/