Noncrossing Sets and a Grassmann Associahedron
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Two Poset Polytopes
Discrete Comput Geom 1:9-23 (1986) G eometrv)i.~.reh, ~ ( :*mllmlati~ml © l~fi $1~ter-Vtrlq New Yorklu¢. t¢ Two Poset Polytopes Richard P. Stanley* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. Two convex polytopes, called the order polytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart poly- nomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer map" then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities. I. The Order Polytope Our aim is to investigate two convex polytopes associated with a finite partially ordered set (poset) P. The first of these, which we call the "order polytope" and denote by O(P), has been the subject of considerable scrutiny, both explicit and implicit, Much of what we say about the order polytope will be essentially a review of well-known results, albeit ones scattered throughout the literature, sometimes in a rather obscure form. The second polytope, called the "chain polytope" and denoted if(P), seems never to have been previously considered per se. It is a special case of the vertex-packing polytope of a graph (see Section 2) but has many special properties not in general valid or meaningful for graphs. -
Double Posets and Real Invariant Varieties Dissertation
Fachbereich Mathematik und Informatik der Freien Universität Berlin Double posets and real invariant varieties Two interactions between combinatorics and geometry Dissertation eingereicht von Tobias Friedl Berlin 2017 Advisor and first reviewer: Prof. Dr. Raman Sanyal Second reviewer: Prof. Dr. Francisco Santos Third reviewer: Priv.-Doz. Dr. Christian Stump Date of the defense: May 19, 2017 Acknowledgements My deepest thanks go to my advisor Raman Sanyal. A PhD-student can only hope for an advisor who is as dedicated and enthusiastic about mathematics as you are. Thank you for getting your hands dirty and spending many hours in front of the blackboard teaching me how to do research. I spent the last years in the amazing and inspiring work group directed by Günter Ziegler at FU Berlin. Thank you for providing such a welcoming and challenging work environment. I really enjoyed the time with my friends and colleagues at the "villa", most of all Francesco Grande, Katharina Jochemko, Katy Beeler, Albert Haase, Lauri Loiskekoski, Philip Brinkmann, Nevena Palić and Jean-Philippe Labbé. Thanks also to Elke Pose, who provides valuable support and keeps bureaucracy at a low level. I want to thank my coauthor Cordian Riener for many fruitful mathematical discussions and for an interesting and enjoyable week of research in Helsinki at Aalto University. Moreover, I’d like to express my gratitude towards my coauthor and friend Tom Chappell. Thanks to the reviewers Paco Santos and Christian Stump for their helpful com- ments and to Christian additionally for many discussions regarding the combinatorics of reflection groups and posets throughout the last years. -
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), . -
Arxiv:1902.07301V3 [Math.CO] 13 Feb 2020 Which N H Itiuinwhere Distribution the and of Ojcuefo 8]Wihhsntbe Eovd H Oere Sole the Resolved
MINUSCULE DOPPELGANGERS,¨ THE COINCIDENTAL DOWN-DEGREE EXPECTATIONS PROPERTY, AND ROWMOTION SAM HOPKINS Abstract. We relate Reiner, Tenner, and Yong’s coincidental down-degree expec- tations (CDE) property of posets to the minuscule doppelg¨anger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelg¨anger pairs behave “as if” they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelg¨anger pairs pretending to have isomor- phic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelg¨anger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type). 1. Introduction Let P be a finite poset. The down-degree of p ∈ P is the number of elements of P which p covers. Consider two probability distributions on P : the uniform distribution; and the distribution where p ∈ P occurs proportional to the number of maximal chains of P containing p. We say that P has the coincidental down-degree expectations (CDE) property if the expected value of the down-degree statistic is the same for these two distributions. -
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. -
The $ H^* $-Polynomial of the Order Polytope of the Zig-Zag Poset
THE h∗-POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET JANE IVY COONS AND SETH SULLIVANT Abstract. We describe a family of shellings for the canonical tri- angulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numer- ator of the Ehrhart series of this order polytope in terms of the swap statistic on alternating permutations. 1. Introduction and Preliminaries The zig-zag poset Zn on ground set {z1,...,zn} is the poset with exactly the cover relations z1 < z2 > z3 < z4 > . That is, this partial order satisfies z2i−1 < z2i and z2i > z2i+1 for all i between n−1 1 and ⌊ 2 ⌋. The order polytope of Zn, denoted O(Zn) is the set n of all n-tuples (x1,...,xn) ∈ R that satisfy 0 ≤ xi ≤ 1 for all i and xi ≤ xj whenever zi < zj in Zn. In this paper, we introduce the “swap” permutation statistic on alternating permutations to give a new combinatorial interpretation of the numerator of the Ehrhart series of O(Zn). We began studying this problem in relation to combinatorial proper- ties of the Cavender-Farris-Neyman model with a molecular clock (or CFN-MC model) from mathematical phylogenetics [4]. We were inter- ested in the polytope associated to the toric variety obtained by ap- plying the discrete Fourier transform to the Cavender-Farris-Neyman arXiv:1901.07443v2 [math.CO] 1 Apr 2020 model with a molecular clock on a given rooted binary phylogenetic tree. We call this polytope the CFN-MC polytope. -
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). -
The Minkowski Property and Reflexivity of Marked Poset Polytopes
The Minkowski property and reflexivity of marked poset polytopes Xin Fang Ghislain Fourier Mathematical Institute Lehrstuhl B f¨ur Mathematik University of Cologne RWTH Aachen University Cologne, Germany Aachen, Germany [email protected] [email protected] Christoph Pegel Institute for Algebra, Number Theory, and Discrete Mathematics Leibniz University Hannover Hannover, Germany [email protected] Submitted: Sep 3, 2018; Accepted: Jan 14, 2020; Published: Jan 24, 2020 c The authors. Released under the CC BY-ND license (International 4.0). Abstract We provide a Minkowski sum decomposition of marked chain-order polytopes into building blocks associated to elementary markings and thus give an explicit minimal set of generators of an associated semi-group algebra. We proceed by characterizing the reflexive polytopes among marked chain-order polytopes as those with the underlying marked poset being ranked. Mathematics Subject Classifications: 52B20, 06A07 1 Introduction To a given finite poset, Stanley [22] associated two polytopes|the order polytope and the chain polytope, which are lattice polytopes having the same Ehrhart polynomial. When the underlying poset is a distributive lattice, Hibi studied in [15] the geometry of the toric variety associated to the order polytope, nowadays called Hibi varieties. Together with Li, they also initiated the study of the toric variety associated to the chain polytope [17]. The singularities of Hibi varieties arising from Gelfand-Tsetlin degenerations of Grassmann varieties are studied by Brown and Lakshmibai in [5] (see also the references therein). Motivated by the representation theory of complex semi-simple Lie algebras, namely the framework of PBW-degenerations, Ardila, Bliem and Salazar [1] introduced the notion the electronic journal of combinatorics 27(1) (2020), #P1.27 1 of marked order polytopes and marked chain polytopes, defined on marked posets. -
Gelfand-Tsetlin Polytopes: a Story of Flow and Order Polytopes
GELFAND-TSETLIN POLYTOPES: A STORY OF FLOW & ORDER POLYTOPES RICKY INI LIU, KAROLA MESZ´ AROS,´ AND AVERY ST. DIZIER Abstract. Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope GT(λ) is equal to the dimension of the corresponding ir- reducible representation of GL(n). It is well-known that the Gelfand-Tsetlin polytope is a marked order polytope; the authors have recently shown it to be a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes. 1. Introduction n Given a partition λ = (λ1; : : : ; λn) 2 Z≥0, the Gelfand-Tsetlin polytope GT(λ) is the set of all nonnegative triangular arrays x11 x12 ··· x1n x22 x23 ··· x2n ······ xn−1;n−1 xn−1;n xnn such that xin = λi for all 1 ≤ i ≤ n; xi−1;j−1 ≥ xij ≥ xi−1;j for all 1 ≤ i ≤ j ≤ n: The integer points of GT(λ) are in bijection with semistandard Young tableaux of shape λ on the alphabet [n]. Moreover, the integer point transform of GT(λ) projects to the Schur function sλ. The latter beautiful result generalizes to Minkowski sums of Gelfand-Tsetlin polytopes and certain Schubert polynomials as well [4]. In this paper we will be interested in the Gelfand-Tsetlin polytope GT(λ) from a purely discrete geometric point of view: we will explore it as a marked order polytope and as a flow polytope. Ardila et. al introduced marked order polytopes and showed that Gelfand-Tsetlin polytopes are examples of them in [1]; the present authors have recently shown that Gelfand-Tsetlin polytopes are flow polytopes [4]. -
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....................... -
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. -
On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial Kolja Knauer, Leonardo Martínez-Sandoval, Jorge Luis Ramírez Alfonsín
On lattice path matroid polytopes: integer points and Ehrhart polynomial Kolja Knauer, Leonardo Martínez-Sandoval, Jorge Luis Ramírez Alfonsín To cite this version: Kolja Knauer, Leonardo Martínez-Sandoval, Jorge Luis Ramírez Alfonsín. On lattice path matroid polytopes: integer points and Ehrhart polynomial. 2017. hal-01457185 HAL Id: hal-01457185 https://hal.archives-ouvertes.fr/hal-01457185 Preprint submitted on 6 Feb 2017 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. ON LATTICE PATH MATROID POLYTOPES: INTEGER POINTS AND EHRHART POLYNOMIAL KOLJA KNAUER, LEONARDO MART´INEZ-SANDOVAL, AND JORGE LUIS RAM´IREZ ALFONS´IN Abstract. In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a charac- terization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid poly- topes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et. al. and present an explicit formula of the Ehrhart polynomial for one of them. 1. Introduction For general background on matroids we refer the reader to [20, 27].