Weak Bruhat Order Research Articles

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

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 $B$.

Chains of maximum length in the Tamari lattice

The Tamari lattice T[subscript n] was originally defined on bracketings of a set of n + 1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, “The enumeration of such paths in Tamari lattices remains mysterious.” The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently, Bergeron and Pr eville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice.

Bruhat order on plane posets and applications

A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal unital bialgebra structure is defined on the vector space HPP generated by PP, using the notion of biideals of plane posets.We here define a partial order on PP, making it isomorphic to the set of partitions with the weak Bruhat order. We prove that this order is compatible with both products of PP; moreover, it encodes a nondegenerate Hopf pairing on the infinitesimal unital bialgebra HPP.

Quasisymmetric Schur functions and modules of the $0$-Hecke algebra

We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions. Nous définissons une action $0$-Hecke sur les tableaux de composition, et ensuite nous l’utilisons pour dériver les modules $0$-Hecke dont la caractéristique quasi-symétrique est une fonction de Schur quasi-symétrique. Nous mettons les modules en relation avec l’ordre de Bruhat faible et les utilisons pour dériver une nouvelle base pour les fonctions quasi-symétriques. Nous classons aussi ces modules qui sont tableau-cycliques et aussi indécomposable. Enfin, nous développons une règle de restriction qui reflète le coproduit des fonctions de Schur quasi-symétriques.

Product decompositions of the symmetric group induced by separable permutations

In this paper we consider the rank generating function of a separable permutation π in the weak Bruhat order on the two intervals [id,π] and [π,w0], where w0=n,n−1,…,1. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on [id,π] and [π,w0], leading to the rank-symmetry and unimodality of the two graded posets.

Abstract root systems and intervals in weak Bruhat order

Abstract We present a new axiomatization of what is known as the “root system” of a Coxeter group that does not involve vector spaces or Coxeter groups. We use this to combinatorially characterize the structure of intervals in weak Bruhat order in much the same way that finite distributive lattices are characterized as the lattices of order ideals of partially ordered sets. In fact, the result for distributive lattices follows as a special case.

Acyclic sets of linear orders via the Bruhat orders

We describe Abello’s acyclic sets of linear orders [SIAM J Discr Math 4(1):1–16, 1991] as the permutations visited by commuting equivalence classes of maximal reduced decompositions. This allows us to strengthen Abello’s structural result: we show that acyclic sets arising from this construction are distributive sublattices of the weak Bruhat order. This, in turn, shows that Abello’s acyclic sets are, in fact, the same as Chameni-Nembua’s distributive covering sublattices (S.T.D.C s). Fishburn’s alternating scheme is shown to be a special case of the Abello/Chameni-Nembua acyclic sets. Any acyclic set that arises in this way can be represented by an arrangement of pseudolines, and we use this representation to derive a simple closed form for the cardinality of the alternating scheme. The higher Bruhat orders prove to be a natural mathematical framework for this approach to the acyclic sets problem.

Tamari lattices, forests and Thompson monoids

A connection relating Tamari lattices on symmetric groups regarded as lattices under the weak Bruhat order to the positive monoid P of Thompson group F is presented. Tamari congruence classes correspond to classes of equivalent elements in P . The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine the Tamari classes. In the monoid P these correspond to the lexicographically largest and the lexicographically smallest form, while on the level of permutations they correspond to 132-avoiding and 231-avoiding permutations. Forests appear naturally in both contexts as they are used to model both permutations and elements of the Thompson monoid. The connection is then extended to Tamari orders on partitions of ( ( k − 1 ) n + 2 ) -gons into ( k + 1 ) -gons and Thompson monoids P k , k ≥ 2 .

Weak Bruhat order on the set of faces of the permutohedron and the associahedron

We define a partial order on the set of faces of the permutohedron, which extends the weak Bruhat order of the symmetric group. The restriction of this order gives a partially ordered set structure to the set of faces of the associahedron, which extends the Tamari order defined on the set of planar binary rooted trees. These orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand, and on the vector space spanned by the set of planar rooted trees on the other hand. Our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak Bruhat order and the Tamari order, respectively, obtained by J.-L. Loday and the second author in [J.-L. Loday, M. Ronco, Order structure and the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002) 253–270].

Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices

We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if Δ(L) is the order complex of a rank (r + 1) geometric lattice L, then for all i ≤ r/2 the h-vector of Δ(L) satisfies hi-1 ≤ hi and hi ≤ hr-i. We also obtain several inequalities for the flag h-vector of Δ(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling---Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of hi-1 ≤ hi when i ≤ (2/7)(r + (5/2)).

Mitosis recursion for coefficients of Schubert polynomials

Mitosis is a rule introduced by Knutson and Miller for manipulating subsets of the n× n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) of Fomin and Kirillov for a permutation w∈ S n by downward induction on weak Bruhat order, thereby generating the coefficients of Schubert polynomials of Lascoux and Schützenberger inductively. This note provides a short and purely combinatorial proof of these properties of mitosis.

Order Structure on the Algebra of Permutations and of Planar Binary Trees

Let X n be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n − 1)-dimensional cube. In each case there exists a graded associative product on ⊕ n≥0 K[X n]. We prove that it can be described explicitly by using the weak Bruhat order on S n, the left-to-right order on planar trees, the lexicographic order in the cube case.

A combinatorial derivation of the Poincaré polynomials of the finite irreducible Coxeter groups

(1) The Poincaré polynomials of the finite irreducible Coxeter groups are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. (2) Non-recursive methods for the computation of ‘standard reduced words’ for (signed) permutations are described. The algebraic basis for both (1) and (2) is a simple partition property of the weak Bruhat order of Coxeter groups into isomorphic parts.

The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

Stanley conjectured that the number of maximal chains in the weak Bruhat order of $S_n$, or equivalently the number of reduced decompositions of the reverse of the identity permutation $ w_0 = n,n-1,n-2,\ldots,2,1$, equals the number of standard Young tableaux of staircase shape $s=\{n-1,n-2,\ldots,1\}$. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraically, later Edelman and Greene found a bijective proof. We provide an extension of the Edelman and Greene bijection to a larger class of words. This extension is similar to the extension of the Robinson-Schensted correspondence to two line arrays. Our proof is inspired by Viennot's planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.

Visibility graphs of staircase polygons and the weak Bruhat order, I: From visibility graphs to maximal chains

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerated configuration ofn points can be associated with amaximal chain in the weak Bruhat order of the symmetric groupS n . The visibility graph ofany simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains andbalanced tableaux of a certain shape. In the case of staircase polygons (special convex fans), we define a class of graphs calledpersistent graphs and show that the visibility graph of a staircase polygon is persistent. We then describe a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs. The question of recovering a staircase polygon from a given persistent graph, via a maximal chain, is studied in the companion paper [4]. The overall goal of both papers is to offer a characterization of visibility graphs, of convex fans.

A Generalization of Rota′s NBC Theorem

We generalize Rota′s theorem characterizing the Möbius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak Bruhat order of a finite Coxeter group and the Tamari lattices. We also give a direct proof of the fact that in the geometric case any total order of the atoms can be used. Simple involutions are used in both proofs. Finally, we show how involutions can be used in similar situations, specifically in a special case of Rota′s Crosscut Theorem as well as in related proofs of Walker on Hall′s Theorem and Reiner on characteristic and Poincaré polynomials.

A Finite Coxeter Group the Weak Bruhat Order of which is not Symmetric Chain

We present a finite irreducible Coxeter group the weak Bruhat order which is not symmetric chain partitionable. This answers one of the questions asked in Björner [1].

Planarity Properties of the Schensted Correspondence

We give a natural decomposition of the set of standard Young tableaux of a given shape into intervals with respect to the weak Bruhat order; each class is completely determined by a partial order on letters which admits a remarkable planar representation.

Finite pseudocomplemented lattices and ‘permutoedre’

We study finite pseudocomplemented lattices and especially those that are also complemented. With regard to the classical results on arbitrary or distributive pseudocomplemented lattices (Glivenko, Stone, Birkhoff, Frink, Grätzer, Balbes, Horn, Varlet,…), the finiteness property allows one to bring significant, more precise, details on the structural properties of such lattices. These results can especially be applied to the lattices defined by the ‘weak Bruhat order’ on a Coxeter group (and, for instance, to the lattice of permutations, called, in french, ‘le treillis permutoèdre’) and to the lattice of binary bracketings.

The Weak Bruhat Order of $\text{S}_\Sigma $, Consistent Sets, and Catalan Numbers

Chains in the weak Bruhat order$\beta $ of $\text{S}_\Sigma $ (the symmetric group on $\Sigma $) belong to the class of subsets of $\text{S}_\Sigma $ over which unrestricted choice necessarily produces transitive relations under pairwise simple majority vote (consistent sets). If for $\text{A} \subset \text{S}_\Sigma $ we let $\text{T}( \text{A} ) \equiv \cup _{\text{p} \in \text{A}} \text{T} ( \text{p} )$ where $\text{T}( \text{p} ) = \{ ( \text{p}_{\text{i}} , \text{p}_{\text{j}} ,\text{p}_{\text{k}} )| \text{i} < \text{j} < \text{k} \}$ and $\Psi ( \text{A} ) \equiv \{ \text{w} \in \text{S}_\Sigma \mid \text{T} ( \text{w} ) \subset \text{T} ( \text{A} ) \}$ the following theorem (among others) is obtained. Theorem. For all${\text{q}} \in {\text{S}}_\Sigma $, if${\text{A}}$is a saturated chain under$\beta $ then $\Psi ( {\text{qA}} )$is an upper semimodular sublattice of cardinality$|\Psi ( {{\text{qA}}} )|\leqq \frac{1}{{|\Sigma | + 1}} (\begin{smallmatrix} 2|\Sigma | \\ |\Sigma | \end{smallmatrix}) \equiv $The$|\Sigma |$th Catalan number. From the Arrow’s Impossibility Theorem point of view, the results obtained here indicate that majority rule produces transitive results if the collection of voters as a whole can be partitioned into no more than $( |\Sigma |^2 + |\Sigma | )/2$ groups which can be ordered according to the level of disagreement they have with respect to a fixed permutation ${\text{p}}$. On the other hand, by viewing ${\text{S}}_\Sigma $ as a Coxeter group a “novel” combinatorial interpretation of the collection of maximal chains that can be obtained from one another by using only one type of Coxeter transformation is obtained.