scholarly journals The Rise-Contact Involution on Tamari Intervals

10.37236/7698 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Viviane Pons
Keyword(s):  
Tamari Lattice

We describe an involution on Tamari intervals and $m$-Tamari intervals. This involution switches two sets of statistics known as the "rises" and the "contacts" and so proves an open conjecture from Préville-Ratelle on intervals of the $m$-Tamari lattice.

scholarly journals A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Wenjie Fang ◽  
Henri Mühle ◽  
Jean-Christophe Novelli
Keyword(s):  
Symmetric Group ◽  
Simple Proof ◽  
Open Problem ◽  
Lattice Isomorphism ◽  
Tamari Lattice ◽  
Lehmer Code

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.

scholarly journals From the Tamari Lattice to Cambrian Lattices and Beyond

2012 ◽  
pp. 293-322 ◽  
Author(s):  
Nathan Reading
Keyword(s):  
Tamari Lattice

scholarly journals A Motzkin filter in the Tamari lattice

2015 ◽  
Vol 338 (8) ◽  
pp. 1370-1378
Author(s):  
Jean-Luc Baril ◽  
Jean-Marcel Pallo
Keyword(s):  
Tamari Lattice

The canonical join complex of the Tamari lattice

2020 ◽  
Vol 174 ◽  
pp. 105207
Author(s):  
Emily Barnard
Keyword(s):  
Tamari Lattice

scholarly journals On a Subposet of the Tamari Lattice

Order ◽  
2013 ◽  
Vol 31 (3) ◽  
pp. 337-363 ◽  
Author(s):  
Sebastian A. Csar ◽  
Rik Sengupta ◽  
Warut Suksompong
Keyword(s):  
Tamari Lattice

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 Balanced binary trees in the Tamari lattice

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Functional Equation ◽  
Binary Trees ◽  
Tamari Lattice ◽  
Tree Patterns ◽  
Generating Series ◽  
Nous Montrons ◽  
International Audience ◽  
Balanced Tree

International audience We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals. Nous montrons que l'ensemble des arbres équilibrés est clos par intervalle dans le treillis de Tamari. Nous caractérisons la forme des intervalles du type $[T_0, T_1]$ où $T_0$ et $T_1$ sont équilibrés en montrant qu'en tant qu'ensembles partiellement ordonnés, ils sont isomorphes à un hypercube. Nous introduisons la notion de motif d'arbre et de grammaire synchrone dans le but d'établir une équation fonctionnelle de la série génératrice qui dénombre les intervalles d'arbres équilibrés.

scholarly journals Extending the Tamari Lattice to Some Compositions of Species

2012 ◽  
pp. 187-210 ◽  
Author(s):  
Stefan Forcey
Keyword(s):  
Tamari Lattice
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