Abstract
We present a generalization of the Tamari lattice to parabolic quotients of the symmetric group. More precisely, we generalize the notions of 231-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients, and show bijectively that these sets are equinumerous. Furthermore, the restriction of weak order on the parabolic quotient to the parabolic 231-avoiding permutations is a lattice quotient. Lastly, we suggest how to extend these constructions to all Coxeter groups. Nous présentons une généralisation du treillis de Tamari aux quotients paraboliques du groupe symétrique. Plus précisément, nous généralisons les notions de permutations qui évitent le motif 231, les partitions non-croisées, et les partitions non-emboîtées aux quotients paraboliques, et nous montrons de façon bijective que ces ensembles sont équipotents. En restreignant l’ordre faible du quotient parabolique aux permutations paraboliques qui évitent le motif 231, on obtient un quotient de treillis d’ordre faible. Enfin, nous suggérons comment étendre ces constructions à tous les groupes de Coxeter.
Highlights
The Tamari lattice T n was introduced by D
Reading’s definition of the Cambrian lattices, which may be described as the restriction of the weak order of a finite Coxeter group to certain aligned elements, which are characterized via their inversion sets [8]
We propose a new generalization of the Tamari lattice to parabolic quotients SJn of the symmetric group by introducing a distinguished subset of permutations, denoted by SJn(231), that play the role of the 231-avoiding permutations for the parabolic quotient
Summary
The Tamari lattice T n was introduced by D. 1365–8050 c 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France In this abstract, we propose a new generalization of the Tamari lattice to parabolic quotients SJn of the symmetric group by introducing a distinguished subset of permutations, denoted by SJn(231), that play the role of the 231-avoiding permutations for the parabolic quotient. We propose a new generalization of the Tamari lattice to parabolic quotients SJn of the symmetric group by introducing a distinguished subset of permutations, denoted by SJn(231), that play the role of the 231-avoiding permutations for the parabolic quotient We characterize these permutations both in terms of a generalized notion of pattern avoidance, and—in our opinion, more naturally—in terms of their inversion sets.
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