A Type-B Associahedron
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SIGNED TREE ASSOCIAHEDRA Vincent PILAUD (CNRS & LIX)
SIGNED TREE ASSOCIAHEDRA Vincent PILAUD (CNRS & LIX) 1 2 3 4 4 2 1 3 POLYTOPES FROM COMBINATORICS POLYTOPES & COMBINATORICS polytope = convex hull of a finite set of Rd = bounded intersection of finitely many half-spaces face = intersection with a supporting hyperplane face lattice = all the faces with their inclusion relations Given a set of points, determine the face lattice of its convex hull. Given a lattice, is there a polytope which realizes it? PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g \ ≥ 321 = H \ H (J) ?6=J([n+1] 312 231 x1=x3 213 132 x1=x2 123 x2=x3 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ = \ H≥(J) 2211 4312 H 3421 6=J [n+1] 3412 ? ( 4213 k-faces of Perm(n) 2212 ≡ ordered partitions of [n + 1] 2431 1211 into n + 1 − k parts 2413 ≡ surjections from [n + 1] 2341 3214 to [n + 1 − k] 1221 1432 2314 1423 1212 3124 1342 1222 1112 1324 2134 1243 1234 PERMUTAHEDRON Permutohedron Perm(n) = conv f(σ(1); : : : ; σ(n + 1)) j σ 2 Σn+1g 4321 \ = \ H≥(J) 2211 4312 H 3421 6=J [n+1] 3412 ? ( 4213 k-faces of Perm(n) 2212 ≡ ordered partitions of [n + 1] 2431 1211 into n + 1 − k parts 2413 ≡ surjections from [n + 1] 2341 3214 to [n + 1 − k] 1221 1432 2314 1423 1212 3124 connections to • inversion sets 1342 1222 1112 • weak order 1324 • reduced expressions 2134 1243 • braid moves 1234 • cosets of the symmetric group ASSOCIAHEDRA ASSOCIAHEDRON Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion vertices $ triangulations vertices $ binary trees edges $ flips edges $ rotations faces $ dissections faces $ Schr¨oder trees VARIOUS ASSOCIAHEDRA Associahedron = polytope whose face lattice is isomorphic to the lattice of crossing-free sets of internal diagonals of a convex (n + 3)-gon, ordered by reverse inclusion (Pictures by Ceballos-Santos-Ziegler) Lee ('89), Gel'fand-Kapranov-Zelevinski ('94), Billera-Filliman-Sturmfels ('90), . -
On a Remark of Loday About the Associahedron and Algebraic K-Theory
An. S¸t. Univ. Ovidius Constant¸a Vol. 16(1), 2008, 5–18 On a remark of Loday about the Associahedron and Algebraic K-Theory Gefry Barad Abstract In his 2006 Cyclic Homology Course from Poland, J.L. Loday stated that the edges of the associahedron of any dimension can be labelled by elements of the Steinberg Group such that any 2-dimensional face represents a relation in the Steinberg Group. We prove his statement. We define a new group R(n) relevant in the study of the rotation distance between rooted planar binary trees . 1 Introduction 1.1 Combinatorics: binary rooted trees and the associahedron Our primary objects of study are planar rooted binary trees. By definition, they are connected graphs without cycles, with n trivalent vertices and n+2 univalent vertices, one of them being marked as the root. 1 2n There are n+1 n binary trees with n internal vertices. Key Words: Rooted binary trees; Associahedron; Algebraic K-Theory, Combinatorics. Mathematics Subject Classification: 05-99; 19Cxx; 19M05; 05C05 Received: October, 2007 Accepted: January, 2008 5 6 Gefry Barad Figure 1. The associahedron Kn is an (n − 1)-dimensional convex polytope whose vertices are labelled by rooted planar binary trees with n internal vertices. Two labelled vertices are connected by an edge if and only if the corresponding trees are connected by an elementary move called rotation. Two trees are connected by a rotation if they are identical, except the zones from the figure below, where one edge is moved into another position; we erase an internal edge and one internal vertex and we glue it again in a different location, to get a rooted binary tree. -
Operations on Partially Ordered Sets and Rational Identities of Type a Adrien Boussicault
Operations on partially ordered sets and rational identities of type A Adrien Boussicault To cite this version: Adrien Boussicault. Operations on partially ordered sets and rational identities of type A. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 2 (2), pp.13–32. hal- 00980747 HAL Id: hal-00980747 https://hal.inria.fr/hal-00980747 Submitted on 18 Apr 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 15:2, 2013, 13–32 Operations on partially ordered sets and rational identities of type A Adrien Boussicault Institut Gaspard Monge, Universite´ Paris-Est, Marne-la-Valle,´ France received 13th February 2009, revised 1st April 2013, accepted 2nd April 2013. − −1 We consider the family of rational functions ψw = Q(xwi xwi+1 ) indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P . In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP . We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P , and we compute its numerator in some special cases. -
Associahedron
Associahedron Jean-Louis Loday CNRS, Strasbourg April 23rd, 2005 Clay Mathematics Institute 2005 Colloquium Series Polytopes 7T 4 × 7TTTT 4 ×× 77 TT 44 × 7 simplex 4 ×× 77 44 × 7 ··· 4 ×× 77 ×× 7 o o ooo ooo cube o o ··· oo ooo oo?? ooo ?? ? ?? ? ?? OOO OO ooooOOO oooooo OO? ?? oOO ooo OOO o OOO oo?? permutohedron OOooo ? ··· OO o ? O O OOO ooo ?? OOO o OOO ?? oo OO ?ooo n = 2 3 ··· Permutohedron := convex hull of (n+1)! points n+1 (σ(1), . , σ(n + 1)) ∈ R Parenthesizing X= topological space with product (a, b) 7→ ab Not associative but associative up to homo- topy (ab)c • ) • a(bc) With four elements: ((ab)c)d H jjj HH jjjj HH jjjj HH tjj HH (a(bc))d HH HH HH HH H$ vv (ab)(cd) vv vv vv vv (( ) ) TTT vv a bc d TTTT vv TTT vv TTTz*vv a(b(cd)) We suppose that there is a homotopy between the two composite paths, and so on. Jim Stasheff Staheff’s result (1963): There exists a cellular complex such that – vertices in bijection with the parenthesizings – edges in bijection with the homotopies – 2-cells in bijection with homotopies of com- posite homotopies – etc, and which is homeomorphic to a ball. Problem: construct explicitely the Stasheff com- plex in any dimension. oo?? ooo ?? ?? ?? • OOO OO n = 0 1 2 3 Planar binary trees (see R. Stanley’s notes p. 189) Planar binary trees with n + 1 leaves, that is n internal vertices: n o ? ?? ? ?? ?? ?? Y0 = { | } ,Y1 = ,Y2 = ? , ? , ( ) ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ? Y3 = ? , ? , ? , ? , ? . Bijection between planar binary trees and paren- thesizings: x0 x1 x2 x3 x4 RRR ll ll RRR ll RRlRll lll RRlll RRR lll lll lRRR lll RRR lll RRlll (((x0x1)x2)(x3x4)) The notion of grafting s t 3 33 t ∨ s = 3 Associahedron n To t ∈ Yn we associate M(t) ∈ R : n M(t) := (a1b1, ··· , aibi, ··· , anbn) ∈ R ai = # leaves on the left side of the ith vertex bi = # leaves on the right side of the ith vertex Examples: ? ? ?? ?? ?? ?? M( ) = (1),M ? = (1, 2),M ? = (2, 1), ? ? ? ?? ?? ?? ?? M ? = (1, 2, 3),M ? = (1, 4, 1). -
Some Lattices of Closure Systems on a Finite Set Nathalie Caspard, Bernard Monjardet
Some lattices of closure systems on a finite set Nathalie Caspard, Bernard Monjardet To cite this version: Nathalie Caspard, Bernard Monjardet. Some lattices of closure systems on a finite set. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2004, 6 (2), pp.163-190. hal-00959003 HAL Id: hal-00959003 https://hal.inria.fr/hal-00959003 Submitted on 13 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 6, 2004, 163–190 Some lattices of closure systems on a finite set Nathalie Caspard1 and Bernard Monjardet2 1 LACL, Universite´ Paris 12 Val-de-Marne, 61 avenue du Gen´ eral´ de Gaulle, 94010 Creteil´ cedex, France. E-mail: [email protected] 2 CERMSEM, Maison des Sciences Economiques,´ Universite´ Paris 1, 106-112, bd de l’Hopital,ˆ 75647 Paris Cedex 13, France. E-mail: [email protected] received May 2003, accepted Feb 2004. In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set we determine the covering relation ≺ of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C ′ where C and C ′ are two such closure systems satisfying C ≺ C ′. -
University of California Santa Cruz
UNIVERSITY OF CALIFORNIA SANTA CRUZ ONLINE LEARNING OF COMBINATORIAL OBJECTS A thesis submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in COMPUTER SCIENCE by Holakou Rahmanian September 2018 The Dissertation of Holakou Rahmanian is approved: Professor Manfred K. Warmuth, Chair Professor S.V.N. Vishwanathan Professor David P. Helmbold Lori Kletzer Vice Provost and Dean of Graduate Studies Copyright © by Holakou Rahmanian 2018 Contents List of Figures vi List of Tables vii Abstract viii Dedication ix Acknowledgmentsx 1 Introduction1 1.1 The Basics of Online Learning.......................2 1.2 Learning Combinatorial Objects......................4 1.2.1 Challenges in Learning Combinatorial Objects..........6 1.3 Background..................................8 1.3.1 Expanded Hedge (EH)........................8 1.3.2 Follow The Perturbed Leader (FPL)................9 1.3.3 Component Hedge (CH).......................9 1.4 Overview of Chapters............................ 11 1.4.1 Overview of Chapter2: Online Learning with Extended Formu- lations................................. 11 1.4.2 Overview of Chapter3: Online Dynamic Programming...... 12 1.4.3 Overview of Chapter4: Online Non-Additive Path Learning... 13 2 Online Learning with Extended Formulations 14 2.1 Background.................................. 20 2.2 The Method.................................. 22 2.2.1 XF-Hedge Algorithm......................... 23 2.2.2 Regret Bounds............................ 26 2.3 XF-Hedge Examples Using Reflection Relations.............. 29 2.4 Fast Prediction with Reflection Relations................. 36 2.5 Projection with Reflection Relations.................... 39 iii 2.5.1 Projection onto Each Constraint in XF-Hedge........... 41 2.5.2 Additional Loss with Approximate Projection in XF-Hedge... 42 2.5.3 Running Time............................ 47 2.6 Conclusion and Future Work....................... -
Single-Element Extensions of Matroids
JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 69B, Nos. 1 and 2, January-June 1965 Single-Element Extensions of Matroids Henry H. Crapo Assistant Professor of Mathematics, Northeastern University, Boston, Mass. (No ve mbe r 16, 1964) Extensions of matroids to sets containing one additional ele ment are c ha rac te ri zed in te rms or modular cuts of the lattice or closed s ubsets. An equivalent characterizati o n is given in terms or linear subclasse s or the set or circuits or bonds or the matroid. A scheme for the construc ti o n or finit e geo· me tric lattices is deri ved a nd th e existe nce of at least 2" no ni somorphic matroids on an II- ele ment set is established. Contents Page 1. introducti on .. .. 55 2. Differentials .. 55 3. Unit in crease runcti ons ... .... .. .... ........................ 57 4. Exact diffe re ntia ls .... ... .. .. .. ........ " . 58 5. Sin gle·ele me nt exte ns ions ... ... 59 6. Linear subclasses.. .. .. ... .. 60 7. Geo metric la ttices ................... 62 8. Numeri ca l c lassification of matroid s .... 62 9. Bibl iography.. .. ............... 64 1. Introduction I of ele ments Pi of a lattice L, in whic h Pi covers Pi - t for i = 1, . .. , n, is a path (of length n) from x to y. In order to facilitate induc tive proofs of matroid 2 theore ms, we shall set forth a characte rization of single A step is a path of length J. element extensions of matroid structures. -
Associahedra Via Spines
ASSOCIAHEDRA VIA SPINES CARSTEN LANGE AND VINCENT PILAUD Abstract. An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches. keywords. Associahedra, polytopal realizations, generalized permutahedra. Associahedra were originally defined as combinatorial objects by J. Stasheff in [Sta63], and later realized as convex polytopes through several different geometric constructions [Lee89, GKZ08, BFS90, Lod04, HL07, PS12, CSZ15], reflecting deep connections to a great variety of mathemat- ical disciplines. They belong to the world of Catalan combinatorics, which can be described by 1 2n+2 several different equivalent models, all counted by the Catalan number Cn+1 = n+2 n+1 , such as parenthesizings of a non-associative product, Dyck paths, binary trees, triangulations, non- crossing partitions, etc [Sta99, Exercice 6.19]. Among the plethora of different approaches, the following description suits best our purposes. An n-dimensional associahedron is a simple polytope such that the inclusion poset of its non-empty faces is isomorphic to the reverse inclusion poset of the dissections of a convex (n + 3)-gon P (i.e. -
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . -
Associahedra, Cyclohedra and a Topological Solution to the A∞ Deligne Conjecture
Advances in Mathematics 223 (2010) 2166–2199 www.elsevier.com/locate/aim Associahedra, cyclohedra and a topological solution to the A∞ Deligne conjecture Ralph M. Kaufmann a,∗, R. Schwell b a Purdue University, Department of Mathematics, 150 N. University St., West Lafayette, IN 47907-2067, United States b Central Connecticut State University, Department of Mathematical Sciences, 1615 Stanley Street, New Britain, CT 06050, United States Received 4 January 2008; accepted 2 November 2009 Available online 4 December 2009 Communicated by Mark Hovey Abstract We give a topological solution to the A∞ Deligne conjecture using associahedra and cyclohedra. To this end, we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally act on the Hochschild cochains of an A∞ algebra yielding an explicit, topological and minimal solution to the A∞ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow-up construction. © 2009 Elsevier Inc. All rights reserved. Keywords: Operads; Deligne’s conjecture; Actions on Hochschild complexes; Homotopy algebras; Polytopes; Cyclohedra; Associahedra; Little discs 0. Introduction In the last years Deligne’s conjecture has been a continued source of inspiration. The original conjecture states that there is a chain model of the little discs operad that acts on the Hochschild cochains of an associative algebra, which induces the known Gerstenhaber structure [9] on coho- * Corresponding author. -
Lattice Thesis
Efficient Representation and Encoding of Distributive Lattices by Corwin Sinnamon A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2018 c Corwin Sinnamon 2018 This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Statement of Contributions This thesis is based on joint work ([18]) with Ian Munro which appeared in the proceedings of the ACM-SIAM Symposium of Discrete Algorithms (SODA) 2018. I contributed many of the important ideas in this work and wrote the majority of the paper. iii Abstract This thesis presents two new representations of distributive lattices with an eye towards efficiency in both time and space. Distributive lattices are a well-known class of partially- ordered sets having two natural operations called meet and join. Improving on all previous results, we develop an efficient data structure for distributive lattices that supports meet and join operations in O(log n) time, where n is the size of the lattice. The structure occupies O(n log n) bits of space, which is as compact as any known data structure and within a logarithmic factor of the information-theoretic lower bound by enumeration. The second representation is a bitstring encoding of a distributive lattice that uses approximately 1:26n bits. -
Math 7409 Lecture Notes 10 Posets and Lattices a Partial Order on a Set
Math 7409 Lecture Notes 10 Posets and Lattices A partial order on a set X is a relation on X which is reflexive, antisymmetric and transitive. A set with a partial order is called a poset. If in a poset x < y and there is no z so that x < z < y, then we say that y covers x (or sometimes that x is an immediate predecessor of y). Hasse diagrams of posets are graphs of the covering relation with all arrows pointed down. A total order (or linear order) is a partial order in which every two elements are comparable. This is equivalent to satisfying the law of trichotomy. A maximal element of a poset is an element x such that if x ≤ y then y = x. Note that there may be more than one maximal element. Lemma: Any (non-empty) finite poset contains a maximal element. In a poset, z is a lower bound of x and y if z ≤ x and z ≤ y. A greatest lower bound (glb) of x and y is a maximal element of the set of lower bounds. By the lemma, if two elements of finite poset have a lower bound then they have a greatest lower bound, but it may not be unique. Upper bounds and least upper bounds are defined similarly. [Note that there are alternate definitions of glb, and in some versions a glb is unique.] A (finite) lattice is a poset in which each pair of elements has a unique greatest lower bound and a unique least upper bound.