UC Berkeley UC Berkeley Electronic Theses and Dissertations

Total Page:16

File Type:pdf, Size:1020Kb

UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title The Generalized External Order, and Applications to Zonotopal Algebra Permalink https://escholarship.org/uc/item/8rx504s8 Author Gillespie, Bryan Rae Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California The Generalized External Order, and Applications to Zonotopal Algebra by Bryan R. Gillespie A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Olga Holtz, Chair Professor Bernd Sturmfels Professor Lauren Williams Professor David Aldous Summer 2018 The Generalized External Order, and Applications to Zonotopal Algebra Copyright 2018 by Bryan R. Gillespie 1 Abstract The Generalized External Order, and Applications to Zonotopal Algebra by Bryan R. Gillespie Doctor of Philosophy in Mathematics University of California, Berkeley Professor Olga Holtz, Chair Extrapolating from the work of Las Vergnas on the external active order for matroid bases, and inspired by the structure of Lenz's forward exchange matroids in the theory of zonotopal algebra, we develop the combinatorial theory of the generalized external order. This partial ordering on the independent sets of an ordered matroid is a supersolvable join- distributive lattice which is a refinement of the geometric lattice of flats, and is fundamentally derived from the classical notion of matroid activity. We uniquely classify the lattices which occur as the external order of an ordered matroid, and we explore the intricate structure of the lattice's downward covering relations, as well as its behavior under deletion and contraction of the underlying matroid. We then apply this theory to improve our understanding of certain constructions in zonotopal algebra. We first explain the fundamental link between zonotopal algebra and the external order by characterizing Lenz's forward exchange matroids in terms of the external order. Next we describe the behavior of Lenz's zonotopal D-basis polynomials under taking directional derivatives, and we use this understanding to provide a new algebraic construction for these polynomials. The construction in particular provides the first known algorithm for computing these polynomials which is computationally tractible for inputs of moderate size. Finally, we provide an explicit construction for the zonotopal P-basis polynomials for the internal and semi-internal settings. i To my parents, for fanning my passions and giving me a foundation on which to build. To my wife, for supporting me through this and filling my days with love and joy. ii Contents Contents ii 1 Introduction 1 2 Background 8 2.1 Matroids and Antimatroids . 8 2.2 Zonotopal Algebra . 19 3 The Generalized External Order 30 3.1 Feasible and Independent Sets of Join-distributive Lattices . 34 3.2 Definition and Fundamental Properties . 37 3.3 Lattice Theoretic Classification . 45 3.4 Deletion and Contraction . 54 3.5 Passive Exchanges and Downward Covering Relations . 57 4 Applications to Zonotopal Algebra 68 4.1 Forward Exchange Matroids and the External Order . 69 4.2 Canonical Basis Polynomials . 71 4.3 Differential Structure of Central D-Polynomials . 77 4.4 Recursive Construction for the Central D-Basis . 79 4.5 Explicit Construction for the Internal and Semi-internal P-Bases . 85 Bibliography 93 A Software Implementations 96 A.1 Ordered Matroids and the External Order . 96 A.2 Polynomial Vector Spaces . 99 A.3 Zonotopal Spaces . 100 iii Acknowledgments This work could not have been completed without the assistance and support of many individuals, both during graduate school and before it, and I owe a debt of gratitude to everyone who has helped me to reach this point. Thanks first to my parents, whose love and encouragement is responsible for much of who I am today. There is nothing more important in the life of a child than a stable and supportive upbringing, and I can only hope to be as good of a parent someday as you both have been for me. From my earliest years of schooling, I would like to thank two of my grade school math teachers who played a formative role in my education. Thanks to Evan McClintock, my 5th grade teacher, for individual after-school math lessons which put me on track to get the most out of later math curricula. Thanks also to Greg Somers for effectively communicating, in so many small ways, the beauty and elegance of rigorous mathematics. You both have a rare gift in your ability to educate, and I am grateful that my path passed through your classrooms. Among my excellent teachers as an undergraduate at Penn State, thanks especially to Svetlana Katok for encouraging my first awkward steps in theoretical mathematics, and to Omri Sarig for showing me a level of clarity and precision in teaching of which I have yet to see an equal. Thanks to Federico Ardilla, Bernd Sturmfels, Lauren Williams, and Martin Olsson for mathematical and academic advice. Grad school has many ups and downs, and your per- spective and suggestions were invariably helpful in weathering them. Thanks to Matthias Lenz, Sarah Brodsky, Spencer Backman, Jose Samper, and Federico Castillo for helpful conversations and thoughtful insights. Additionally, thanks to Anne Schilling and Nicolas Thi´eryfor a whirlwind initiation into the world of SageMath coding. Thanks especially to my advisor, Olga Holtz, for providing the starting point for my work and much invaluable guidance, and also for allowing me the freedom to explore the mathematics that excited me the most. This dissertation simply would not have been possible without your input and assistance. Thanks also to Bernd Sturmfels, Lauren Williams, and David Aldous for serving on my dissertation committee. Finally, thank you to my wife Maria, for your unwavering support and encouragement throughout graduate school. Your energy, enthusiasm, and determination are an inspiration to me. I look forward to sharing many more years of mathematical adventures with you. The material in this dissertation is based in part upon work supported by the National Science Foundation under Grant No. DMS-1303298. Thanks to the NSF for the time and freedom this support afforded me over several semesters, and thanks also to the Berkeley Graduate Division for support provided by the Graduate Division Summer Grant. 1 Chapter 1 Introduction The theory of zonotopal algebra studies the characteristics and applications of certain finite- dimensional polynomial vector spaces, and related annihilating ideals, which are derived from the structure of particular combinatorial and geometric objects formed from a finite ordered list of vectors. The theory lies at an interface between several starkly contrasting mathemat- ical disciplines, especially approximation theory, commutative algebra, and matroid theory, and wide-ranging connections have been found with topics in enumerative combinatorics, representation theory, and discrete geometry. Zonotopal algebra has at its foundation a collection of ideas in numerical analysis and approximation theory that were developed in the 1980s and early 90s, in works such as [1, 9, 10, 11, 17, 19, 16]. In these works, the central zonotopal spaces were discovered in relation to the approximation of multivariate functions by so-called (exponential) box splines, and as the solutions of certain classes of multivariate differential and difference equations. The central zonotopal spaces lie in the real polynomial ring Π = R[x1; : : : ; xd], and consist of two finite-dimensional polynomial vector spaces D(X) and P(X), and two related ideals I(X) and J (X) which are constructed from the columns of a d × n matrix X. The D-space, also known as the Dahmen-Michelli space, is the space of polynomials spanned by the local polynomial pieces of the box spline associated with the matrix X. It can be realized as the differential kernel or Macaulay inverse system of the J -ideal, an ideal generated by products of linear forms corresponding with the cocircuits of the linear independence matroid of the columns of X. The P-space is defined using the notion of matroid activity of bases of X, and acts as the dual vector space of the D-space under the identification p 7! hp; ·i, where h·; ·i :Π×Π ! R : denotes the differential bilinear form given by hp; qi = p(@)q x=0, and where p(@) is the differential operator obtained from p by replacing each variable xi with the operator @=@xi. This space is realized as the differential kernel of the I-ideal, a power ideal generated by certain powers of linear forms corresponding to orthogonal vectors of the hyperplanes spanned by column vectors of X. The D- and P-spaces in particular each contain a unique interpolating polynomial for any function defined on the vertex set of certain hyperplane arrangements associated with CHAPTER 1. INTRODUCTION 2 the matrix X, a property known as correctness. In addition, the Dahmen-Michelli space has a simple construction in terms of the least map, an operator introduced in [11] which plays an important role in interpolation theory. Furthermore, relations dual to these can be developed with respect to generic vertex sets for the zonotope of X, which is formed by taking the Minkowski sum of the segments [0; x] = ftx : t 2 [0; 1]g for the column vectors x of X. It is from this geometric connection (and related combinatorial structure) that the name zonotopal algebra is derived. 1 0 1 X = 0 1 1 Figure 1.1: A hyperplane arrangement and corresponding dual zonotope of the matrix X. Particularly in the last decade, the area of zonotopal algebra has witnessed a resurgence of interest, with applications and generalizations of the theory emerging in a variety of new directions. In [38], Sturmfels and Xu study sagbi bases of Cox-Nagata rings, and discuss an algebraic generalization of the zonotopal spaces called zonotopal Cox rings which can be described as subalgebras of particular Cox-Nagata rings.
Recommended publications
  • 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.
    [Show full text]
  • 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 ′.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Doubling Tolerances and Coalition Lattices
    DOUBLING TOLERANCES AND COALITION LATTICES GABOR´ CZEDLI´ Dedicated to the memory of Ivo G. Rosenberg Abstract. If every block of a (compatible) tolerance (relation) T on a mod- ular lattice L of finite length consists of at most two elements, then we call T a doubling tolerance on L. We prove that, in this case, L and T determine a modular lattice of size 2jLj. This construction preserves distributivity and modularity. In order to give an application of the new construct, let P be a partially ordered set (poset). Following a 1995 paper by G. Poll´akand the present author, the subsets of P are called the coalitions of P . For coalitions X and Y of P , let X ≤ Y mean that there exists an injective map f from X to Y such that x ≤ f(x) for every x 2 X. If P is a finite chain, then its coalitions form a distributive lattice by the 1995 paper; we give a new proof of the distributivity of this lattice by means of doubling tolerances. 1. Introduction There are two words in the title that are in connection with Ivo G. Rosenberg; these words are \tolerance" and \lattice", both occurring also in the title of our joint lattice theoretical paper [6] (coauthored also by I. Chajda). This fact encouraged me to submit the present paper to a special volume dedicated to Ivo's memory even if this volume does not focus on lattice theory. The paper is structured as follows. In Section 2, after few historical comments on lattice tolerances, we introduce the concept of doubling tolerances (on lattices) as those tolerances whose blocks are at most two-element.
    [Show full text]
  • Ordered Sets an Introduction
    Bernd S. W. Schröder Ordered Sets An Introduction Birkhäuser Boston • Basel • Berlin Bernd S. W. Schröder Program of Mathematics and Statistics Lousiana Tech University Ruston, LA 71272 U.S.A. Library of Congress Cataloging-in-Publication Data Schröder, Bernd S. W. (Bernd Siegfried Walter), 1966- Ordered sets : an introduction / Bernd S. W. Schröder. p. cm. Includes bibliographical references and index. ISBN 0-8176-4128-9 (acid-free paper) – ISBN 3-7643-4128-9 (acid-free paper) 1. Ordered sets. I. Title. QA171.48.S47 2002 511.3'2–dc21 2002018231 CIP AMS Subject Classifications: 06-01, 06-02 Printed on acid-free paper. ® ©2003 Birkhäuser Boston Birkhäuser All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4128-9 SPIN 10721218 ISBN 3-7643-4128-9 Typeset by the author. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 Birkhäuser Boston • Basel • Berlin A member of BertelsmannSpringer Science+Business Media GmbH Contents Table of Contents .
    [Show full text]
  • Noncrossing Partitions, Bruhat Order and the Cluster Complex Philippe Biane, Matthieu Josuat-Vergès
    Noncrossing partitions, Bruhat order and the cluster complex Philippe Biane, Matthieu Josuat-Vergès To cite this version: Philippe Biane, Matthieu Josuat-Vergès. Noncrossing partitions, Bruhat order and the cluster com- plex. Annales de l’Institut Fourier, Association des Annales de l’Institut Fourier, 2019, 69 (5), pp.2241- 2289. 10.5802/aif.3294. hal-01687706 HAL Id: hal-01687706 https://hal.archives-ouvertes.fr/hal-01687706 Submitted on 18 Jan 2018 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. NONCROSSING PARTITIONS, BRUHAT ORDER AND THE CLUSTER COMPLEX PHILIPPE BIANE AND MATTHIEU JOSUAT-VERGES` Abstract. We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular we study the restriction of these orders to noncrossing partitions and show that the intervals for these or- ders can be enumerated in terms of the cluster complex. The properties of our orders permit to revisit several results in Coxeter combinatorics, such as the Chapoton triangles and how they are related, the enumera- tion of reflections with full support, the bijections between clusters and noncrossing partitions.
    [Show full text]
  • Balanced Binary Trees in the Tamari Lattice
    FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 593–604 Balanced binary trees in the Tamari lattice Samuele Giraudo1 1Institut Gaspard Monge, Universite´ Paris-Est Marne-la-Vallee,´ 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee´ cedex 2, France Abstract. We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0;T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals. Resum´ e.´ Nous montrons que l’ensemble des arbres equilibr´ es´ est clos par intervalle dans le treillis de Tamari. Nous caracterisons´ la forme des intervalles du type [T0;T1] ou` T0 et T1 sont equilibr´ es´ en montrant qu’en tant qu’ensembles partiellement ordonnes,´ ils sont isomorphes a` un hypercube. Nous introduisons la notion de motif d’arbre et de gram- maire synchrone dans le but d’etablir´ une equation´ fonctionnelle de la serie´ gen´ eratrice´ qui denombre´ les intervalles d’arbres equilibr´ es.´ Keywords: balanced trees, Tamari lattice, posets, grammars, generating series, combinatorics 1 Introduction Binary search trees are used as data structures to represent dynamic totally ordered sets [7, 6, 3]. The algorithms solving classical related problems such as the insertion, the deletion or the search of a given element can be performed in a time logarithmic in the cardinality of the represented set, provided that the encoding binary tree is balanced. Recall that a binary tree is balanced if for each node x, the height of the left subtree of x and the height of the right subtree of x differ by at most one.
    [Show full text]
  • Arxiv:Math/0609184V2 [Math.CO] 18 May 2007 Atb S Rn M-611.LW a Upre Npr Ya by Part in Supported Was L.W
    FACES OF GENERALIZED PERMUTOHEDRA ALEXANDER POSTNIKOV, VICTOR REINER, AND LAUREN WILLIAMS Abstract. The aim of the paper is to calculate face numbers of simple gen- eralized permutohedra, and study their f-, h- and γ-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zono- topes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving de- scent statistics. This includes a combinatorial interpretation for γ-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal’s conjecture on nonnegativity of γ-vectors. We calculate explicit generating functions and formulae for h-polynomials of various families of graph-associahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb’s problem. We give (and conjecture) upper and lower bounds for f-, h-, and γ-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes. Contents 1. Introduction 2 2. Face numbers 5 2.1. Polytopes, cones, and fans 5 2.2. f-vectors and h-vectors 6 2.3. Flag simple polytopes and γ-vectors 7 3. Generalized permutohedra and the cone-preposet dictionary 8 3.1. Generalized permutohedra 9 3.2. Braid arrangement 9 3.3. Preposets,equivalencerelations,andposets 10 3.4. The dictionary 11 4. Simple generalized permutohedra 14 arXiv:math/0609184v2 [math.CO] 18 May 2007 4.1. Descents of tree-posets and h-vectors 14 4.2. Bounds on the h-vector and monotonicity 15 5.
    [Show full text]
  • On a Subposet of the Tamari Lattice
    FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 567–578 On a Subposet of the Tamari Lattice Sebastian A. Csar1 and Rik Sengupta2 and Warut Suksompong3 1School of Mathematics, University Of Minnesota, Minneapolis, MN 55455, USA 2Department of Mathematics, Princeton University, NJ 08544, USA 3Department of Mathematics, Massachusetts Institute of Technology, MA 02139, USA Abstract. We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989). Resum´ e.´ Nous discutons d’un subposet du treillis de Tamari introduit par Pallo. Nous appellons ce poset le comb poset. Nous montrons que trois fonctions binaires qui ne se comptent pas bien dans le trellis de Tamari se comptent bien dans un intervalle du comb poset : distance dans le trellis de Tamari, le supremum et l’infimum et les parsewords communs. De plus, nous discutons un rapport entre ce poset et un ordre partiel dans le groupe symetrique´ etudi´ e´ par Edelman. Keywords: poset, Tamari lattice, tree rotation 1 Introduction The set Tn of all complete binary trees with n leaves, or, equivalently, parenthesizations of n letters, has been well-studied. Of particular interest here is a partial order on Tn, giving the well-studied Tamari lattice, Tn.
    [Show full text]