Crossings and nestings in set partitions of classical types

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections that interchange crossings and nestings. For types B and C, they generalize a construction by Kasraoui and Zeng for type A, whereas for type D, we were only able to construct a bijection between non-crossing and non-nesting set partitions. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of the maximal crossing with the cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type A. Using a variant of this bijection, we also settle a conjecture by Soll and Welker concerning generalized type B triangulations and symmetric fans of Dyck paths.Comment: 22 pages, 7 Figures, removed erroneous commen

Multi-cluster complexes

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

Subwords and Plane Partitions

International audienceUsing the powerful machinery available for reduced words of type $B$, we demonstrate a bijection between centrally symmetric $k$-triangulations of a $2(n + k)$-gon and plane partitions of height at most $k$ in a square of size $n$. This bijection can be viewed as the type $B$ analogue of a bijection for $k$-triangulations due to L. Serrano and C. Stump.En utilisant la machinerie puissante pour mots réduits de type $B$, nous démontrons une bijection entre les $k$-triangulations centralement symétriques d’un $2(n + k)$-gon et les partitions de plans de hauteur inférieure ou égale à $k$ dans un carré de taille $n$. Cette bijection peut être considérée comme l’analogue de type $B$ d’une bijection de $k$-triangulations due à L. Serrano et C. Stump

New interpretations for noncrossing partitions of classical types

We interpret noncrossing partitions of type $B$ and type $D$ in terms of noncrossing partitions of type $A$. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type $B$, type $C$ and type $D$ which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type $B$ and type $D$, and find bijections between them and noncrossing partitions of type $B$ and type $D$ respectively.Comment: 21 pages, 15 figures, final versio

Arc-Coloured Permutations

The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. otherlanguage*french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. otherlanguage*</end

Crossings and nestings in colored set partitions

Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call \emph{$r$-colored set partitions}). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that $r$-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in \NN^r, generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further revised, additional section adde