scholarly journals Cataland: Why the Fuss?

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Christian Stump ◽  
Hugh Thomas ◽  
Nathan Williams
Keyword(s):  
Coxeter System ◽  
Cluster Complexes ◽  
The Third ◽  
International Audience

International audience The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

scholarly journals Multi-cluster complexes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Jean-Philippe Labbé ◽  
Christian Stump
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Simplicial Complexes ◽  
Cluster Complex ◽  
Geometric Properties ◽  
Cluster Complexes ◽  
Combinatorial Description ◽  
Compatibility Relation ◽  
Positive Roots ◽  
International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

scholarly journals How to visualize high-dimensional data: a roadmap

2020 ◽  
Vol Special issue on... ◽  
Author(s):  
Hermann Moisl
Keyword(s):  
Cluster Analysis ◽  
High Dimensional Data ◽  
Latent Structure ◽  
High Dimensional ◽  
Graphical Methods ◽  
Data Set ◽  
The Third ◽  
Historical Text ◽  
International Audience ◽  
And Cluster Analysis

International audience Discovery of the chronological or geographical distribution of collections of historical text can be more reliable when based on multivariate rather than on univariate data because multivariate data provide a more complete description. Where the data are high-dimensional, however, their complexity can defy analysis using traditional philological methods. The first step in dealing with such data is to visualize it using graphical methods in order to identify any latent structure. If found, such structure facilitates formulation of hypotheses which can be tested using a range of mathematical and statistical methods. Where, however, the dimensionality is greater than 3, direct graphical investigation is impossible. The present discussion presents a roadmap of how this obstacle can be overcome, and is in three main parts: the first part presents some fundamental data concepts, the second describes an example corpus and a high-dimensional data set derived from it, and the third outlines two approaches to visualization of that data set: dimensionality reduction and cluster analysis.

scholarly journals Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type $B$

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Kyoungsuk Park
Keyword(s):  
Bruhat Order ◽  
Type B ◽  
Coxeter System ◽  
Covering Relation ◽  
Signed Permutations ◽  
Weak Bruhat Order ◽  
International Audience

International audience Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$. De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$.

scholarly journals Boolean complexes and boolean numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Coxeter Group ◽  
Bruhat Order ◽  
Graph Invariant ◽  
Coxeter System ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Order Ideals ◽  
Simplicial Poset ◽  
The Ideal

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

scholarly journals On the Barban-Davenport-Halberstam theorem : X

1998 ◽  
Vol Volume 21 ◽  
Author(s):  
C Hooley
Keyword(s):  
Large Sieve ◽  
The Third ◽  
International Audience

International audience We return to the topics of the third and ninth papers of the series, and the main aim is to remove some conditions on the previous results by means of a large-sieve estimate due to H. L. Montgomery.

scholarly journals Unital versions of the higher order peak algebras

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Marcelo Aguiar ◽  
Jean-Christophe Novelli ◽  
Jean-Yves Thibon
Keyword(s):  
Higher Order ◽  
Type B ◽  
The Third ◽  
Nous Montrons ◽  
International Audience

International audience We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411―430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type $B$. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781―2824]. Nous construisons des extensions unitaires des algèbres de pics d'ordre supérieur définies par Krob et le troisième auteur dans [Ann. Comb. 9 (2005), 411―430], et nous montrons qu'elles peuvent être obtenues comme images homomorphes de certaines sous-algèbres des algèbres de Mantaci-Reutenauer de type $B$. Ceci généralise un résultat dû à Bergeron, Nyman et au premier auteur [Trans. AMS 356 (2004), 2781―2824].

scholarly journals Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Anders Claesson ◽  
Mark Dukes ◽  
Sergey Kitaev
Keyword(s):  
Generating Function ◽  
Integer Sequences ◽  
Combinatorial Objects ◽  
New Class ◽  
Chord Diagrams ◽  
The Third ◽  
Finite Series ◽  
International Audience ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

scholarly journals The Sorting Order on a Coxeter Group

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Coxeter Group ◽  
Bruhat Order ◽  
Distributive Lattices ◽  
Coxeter System ◽  
Maximal Lattice ◽  
International Audience ◽  
The Way

International audience Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.

scholarly journals Compatibility fans realizing graphical nested complexes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Thibault Manneville ◽  
Vincent Pilaud
Keyword(s):  
Finite Type ◽  
Type B ◽  
Cluster Complexes ◽  
Explicit Constructions ◽  
International Audience ◽  
Connected Subgraphs ◽  
Generalized Associahedra

International audience Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.

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