Meeting covered elements in ν-Tamari lattices

2022 ◽  
Vol 134 ◽  
pp. 102303
Author(s):  
Colin Defant
Keyword(s):  
Tamari Lattices

scholarly journals Tamari lattices, forests and Thompson monoids

2007 ◽  
Vol 28 (4) ◽  
pp. 1216-1238 ◽  
Author(s):  
Zoran Šunić
Keyword(s):  
Tamari Lattices

scholarly journals Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

2021 ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Structural Investigation ◽  
Canonical Representation ◽  
Weak Order ◽  
Noncrossing Partitions ◽  
Tamari Lattice ◽  
Semidistributive Lattice ◽  
Tamari Lattices

AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

scholarly journals Tamari Lattices for Parabolic Quotients of the Symmetric Group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Henri Mühle ◽  
Nathan Williams
Keyword(s):  
Symmetric Group ◽  
Coxeter Groups ◽  
Weak Order ◽  
Set Partitions ◽  
Tamari Lattice ◽  
Nous Montrons ◽  
International Audience ◽  
Tamari Lattices

International audience 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.

scholarly journals Counting smaller elements in the Tamari and m-Tamari lattices

2015 ◽  
Vol 134 ◽  
pp. 58-97 ◽  
Author(s):  
Grégory Châtel ◽  
Viviane Pons
Keyword(s):  
Tamari Lattices

scholarly journals KP line solitons and Tamari lattices

2010 ◽  
Vol 44 (2) ◽  
pp. 025203 ◽  
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen
Keyword(s):  
Tamari Lattices

scholarly journals The Number of Intervals in the $m$-Tamari Lattices

10.37236/2027 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Éric Fusy ◽  
Louis-François Préville-Ratelle
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Open Problem ◽  
Lattice Structure ◽  
Square Grid ◽  
Tamari Lattice ◽  
Bijective Proof ◽  
Tamari Lattices

An $m$-ballot path of size $n$ is a path on the square grid consisting of north and east steps, starting at $(0,0)$, ending at $(mn,n)$, and never going below the line $\{x=my\}$. The set of these paths can be equipped with a lattice structure, called the $m$-Tamari lattice and denoted by $\mathcal{T}_n^{(m)}$, which generalizes the usual Tamari lattice $\mathcal{T}_n$ obtained when $m=1$. We prove that the number of intervals in this lattice is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This formula was recently conjectured by Bergeron in connection with the study of diagonal coinvariant spaces. The case $m=1$ was proved a few years ago by Chapoton. Our proof is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. The solution of this equation is an algebraic series, obtained by a guess-and-check approach. Finding a bijective proof remains an open problem.

scholarly journals On Tamari lattices

1994 ◽  
Vol 133 (1-3) ◽  
pp. 99-122 ◽  
Author(s):  
Winfried Geyer
Keyword(s):  
Tamari Lattices
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