On permutation lattices

Duquenne, Vincent; Cherfouh, Améziane

doi:10.1016/0165-4896(94)00733-0

Cited by 18 publications

(14 citation statements)

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“…With varying levels of generality, many lattice and order properties of this lattice have been determined. (See, for example, references in [7], [9], [14], [16] and [17].) This paper continues a program, begun in [18], of applying lattice theory to gain new insights into the combinatorics and geometry of Coxeter groups.…”

Section: Introductionmentioning

confidence: 99%

Sortable elements and Cambrian lattices

Reading

2007

Algebra univers.

145

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Abstract. We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the Cambrian lattice is the weak order on Coxetersortable elements. These results exhibit W -Catalan combinatorics arising in the context of the lattice theory of the weak order on W. IntroductionThe weak order on a finite Coxeter group is a lattice [3] which encodes much of the combinatorics and geometry of the Coxeter group. The weak order has been studied in the special case of the permutation lattice and in the broader generality of the poset of regions of a simplicial hyperplane arrangement. With varying levels of generality, many lattice and order properties of this lattice have been determined. (See, for example, references in [7], [9], [14], [16] and [17].) This paper continues a program, begun in [18], of applying lattice theory to gain new insights into the combinatorics and geometry of Coxeter groups. Specifically, we solidify the connection, first explored in [20], between the lattice theory of the weak order and the combinatorics of the W -Catalan numbers. These numbers count, among other things, the vertices of the (simple) generalized associahedron (a polytope which encodes the underlying structure of cluster algebras of finite type [10,11]), the W -noncrossing partitions (which provide an approach [2,6] to the geometric group theory of the Artin group associated to W ) and the sortable elements [21] of W (which we discuss below).2000 Mathematics Subject Classification. 20F55, 06B10. The author was partially supported by NSF grants DMS-0202430 and DMS-0502170. In [20], the Cambrian lattices were defined as lattice quotients of the weak order on W modulo certain congruences, identified as the join (in the lattice of congruences of the weak order) of a small list of join-irreducible congruences. Any lattice quotient of the weak order defines [19] a complete fan which coarsens the fan defined by the reflecting hyperplanes of W, and in [20] it was conjectured that the fan associated to a Cambrian lattice is combinatorially isomorphic to the normal fan of the corresponding generalized associahedron. In particular, each Cambrian lattice was conjectured to have cardinality equal to the W -Catalan number. These conjectures were proved for two infinite families of finite Coxeter groups (A n and B n ).The definition of Coxeter-sortable elements (or simply sortable elements) of W was inspired by the effort to better understand Cambrian lattices. Sortable elements were introduced in [21] and used to give a bijective proof that W -noncrossing partitions are equinumerous with vertices of the generalized associahedron. In this paper, we make explicit the essential connection between sortable elements and Cambrian lattices, proving in particular that the elements of the Cambrian lattice are counted by the W -Catalan number. The conjecture from [20] on the combinatorial ...

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Section: Introductionmentioning

confidence: 99%

Sortable elements and Cambrian lattices

Reading

2007

Algebra univers.

145

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“…In Theorem 2.4, below, we describe canonical join representations of permutations explicitly, in particular giving an independent proof that the weak order on permutations is semidistributive. The argument in [6] is very different, but a proof that is similar in spirit to the proof here can be obtained by combining [15,Proposition 6.4] Figure 3. The weak order on S n for n " 3 and for n " 4…”

Section: Canonical Join Representations Of Permutationsmentioning

confidence: 74%

Noncrossing Arc Diagrams and Canonical Join Representations

Reading¹

2015

SIAM J. Discrete Math.

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We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each permutation x has a canonical join representation: a unique lowest set of permutations joining to x. The second problem is to determine which sets of permutations appear as canonical join representations. The two problems turn out to be closely related because the noncrossing arc diagrams provide a combinatorial model for canonical join representations. The same considerations apply more generally to lattice quotients of the weak order. Considering quotients produces, for example, a new combinatorial object counted by the Baxter numbers and an analogous new object in bijection with generic rectangulations. arXiv:1405.6904v4 [math.CO]

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“…This graph structure has already been studied in the literature and the properties that are indicated below are well known [3,11,6].…”

Section: Basic Propertiesmentioning

confidence: 95%

Characterization of all ρ-approximated sequences for some scheduling problems

Billaut

López

2011

Etfa2011

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Some scheduling problems present the peculiarity to be solvable in polynomial time and to have a huge number of optimal solutions. In the disturbed environment of a production manufacturing system, where the forecasted schedule is going to change because of unexpected events or uncertainties, it can be interesting not only to know one or several optimal sequences, but the characteristics of 'good' sequences. In this paper, we focus on the characterization of all the ρ-approximated sequences, which are solutions of a scheduling problem with a performance not worse than a given distance from the value of the optimal solution.With the support of the lattice of permutations, we define the characteristics of the optimal sequences for some particular scheduling problems. We present a method which is able, for some specific scheduling problems, to give the characteristics of all the ρ-approximated sequences. A computational experience is carried out to evaluate the performance of the proposed method.

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On permutation lattices

Cited by 18 publications

References 5 publications

Sortable elements and Cambrian lattices

Sortable elements and Cambrian lattices

Noncrossing Arc Diagrams and Canonical Join Representations

Characterization of all ρ-approximated sequences for some scheduling problems

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On permutation lattices

Cited by 18 publications

References 5 publications

Sortable elements and Cambrian lattices

Sortable elements and Cambrian lattices

Noncrossing Arc Diagrams and Canonical Join Representations

Characterization of all &#x03C1;-approximated sequences for some scheduling problems

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Characterization of all ρ-approximated sequences for some scheduling problems