Enumeration of Tilings and Related Problems
|
|
- Aleesha Greer
- 5 years ago
- Views:
Transcription
1 Enumeration of Tilings and Related Problems Tri Lai Institute for Mathematics and its Applications Minneapolis, MN Discrete Mathematics Seminar University of British Columbia Vancouver February 2016
2 Outline 1 Introduction to the Enumeration of Tilings. 2 Electrical networks 3 Tiling expression of minors 4 Future work.
3 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions.
4 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge.
5 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps.
6 Definition of Tilings A lattice divides the plane into disjoint pieces, called fundamental regions. A tile is a union of any two fundamental regions sharing an edge. A tiling of a region is a covering of the region by tiles so that there are no gaps or overlaps. The number of tilings of a 8 8 chessboard is 12,988,816.
7 Tilings We would like to find the number of tilings of certain regions.
8 Tilings We would like to find the number of tilings of certain regions. Tilings can carry weights, we also care about the weighted sum of tilings: T wt(t ), called the tiling generating function.
9 Kasteleyn Temperley Fisher Theorem Theorem (Kasteleyn, Temperley and Fisher 1961) The number of tilings of a 2m 2n chessboard equals m n j=1 k=1 ( ( 4 cos 2 jπ ) ( + 4 cos 2 kπ ) ). 2m + 1 2n + 1
10 Tilings and dimer coverings
11 Pfaffian Let A = (a i,j ) be a 2N 2N skew-symmetric matrix. The Pfaffian of A is defined by the equation Pf(A) = 1 2 N N! σ S 2N sign(σ) N i=1 a σ(2i 1),σ(2i) where S 2N is the symmetric group and sign(σ) is the signature of σ.
12 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n.
13 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n. The adjacent matrix of G m,n is A = (a i,j ) 2N 2N (N = mn) defined as: 1 if i j, a i,j = 1 if j i, 0 if {i, j} / E.
14 Pfaffian Let G m,n is an orientation of the grid graph of size 2m 2n. The adjacent matrix of G m,n is A = (a i,j ) 2N 2N (N = mn) defined as: 1 if i j, a i,j = 1 if j i, 0 if {i, j} / E. There exists an orientation so that Pf(A) = M(G m,n ).
15 Pfaffian and Determinant Theorem (Muir) det A = (Pf(A)) 2. (det(a)) 2 = m n j=1 k=1 ( ( 4 cos 2 jπ ) ( + 4 cos 2 kπ ) ) 4 2m + 1 2n + 1
16 Connections and Applications to Other Fields Statistical Mechanics: Dimer model, double-dimer model, square ice model, 6-vertex model, fully packed loops configuration... Probability: Arctic Curves. Graph Theory: Bijection between tilings and perfect matchings, Temperley s correspondence. Cluster Algebras: Combinatorial interpretation of cluster variables. Electrical networks:... Other topics in combinatorics: Alternating sign matrices, monotone triangles, plane partitions, lattice paths, symmetric functions...
17 Aztec Diamond Theorem (Elkies, Kuperberg, Larsen and Propp 1991) The Aztec diamond of order n has 2 n(n+1)/2 domino tilings. Figure: The Aztec diamond of order 5 and one of its tilings.
18 An Aztec Temple
19 Urban renewal Definition M(G) = wt(π) π M(G) Lemma (Spider Lemma) M(G) = (xz + yt) M(G ) Lemma (Vertex-spitting Lemma) M(G) = M(G )
20 Proof using Urban renewal
21 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 In 1999, James Propp collected 32 open problems in enumeration of tilings.
22 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 In 1999, James Propp collected 32 open problems in enumeration of tilings. Problem 16 on the list asks for the number of tilings of a quasi-hexagon.
23 Quasi-hexagon a=5 b=3 c=3 c=3 a=5 b=3 Theorem (L. 2014) The number of tilings of a quasi-hexagon is a power of 2 times the number of tilings of a semi-regular hexagon.
24 MacMahon s Formula Theorem (MacMahon 1900) The number of (lozenge) tilings of a semi-regular hexagon of sides a, b, c, a, b, c on the triangular lattice is a b c i=1 j=1 t=1 i + j + t 1 H(a) H(b) H(c) H(a + b + c) = i + j + t 2 H(a + b) H(b + c) H(c + a), where the hyperfactorial H(n) = 0! 1! 2!... (n 1)!. a=4 b=5 c=6 c=6 b=5 a=4
25 Tilings and non-intersecting lattice paths Bijection between lozenge tilings and non-intersecting lattice paths
26 Lindström-Gessel-Viennot Lemma Lemma G = (E, V ) is an acyclic directed graph, A = {a 1, a 2,..., a k } and B = {b 1, b 2,..., b k } are disjoint subsets of V. Assume that any pair of paths connecting a i and b j, and a j and b i are intersected. The number of families of non-intersecting lattice paths connecting A and B is given by P(A B) = det(p i,j ) k k where p i,j is the number of paths from a i to b j.
27 Connection to Plane Partitions a=10 O j k i b=10 c=10 c=10 b=10 a=10
28 MacMahon s q-formula Theorem (MacMahon) π q vol(π) = H q(a) H q (b) H q (c) H q (a + b + c) H q (a + b) H q (b + c) H q (c + a), where the sum is taken over all stacks π fitting in an a b c box. q-integer [n] q := 1 + q + q q n 1 q-factorial [n] q! := [1] q [2] q [3] q... [n] q q-hyperfactorial H q (n) := [0] q! [1] q! [2] q!... [n 1] q!.
29 Generalizing MacMahon s Formula a z+b+c x m y j O k i a z+b+c x m y t m x c c m a y+b b t+c t m a+b x c a z m q volume of the stack =? stacks c m y+b 3 b 1 m 2 z t+c a+b
30 Generalizing MacMahon s Formula Theorem (L ) For non-negative integers x, y, z, t, m, a, b, c q vol(π) H q( + x + y + z + t) = H q( + x + y + t) H q( + x + y + z) π Hq( + x + t) Hq( + x + y) Hq( + y + z) Hq( ) H q( + z + t) H q( + x) H q( + y) Hq(m + b + c + z + t) Hq(m + a + c + x) Hq(m + a + b + y) H q(m + b + y + z) H q(m + c + x + t) H q(c + x + t) H q(b + y + z) H q(a + c + x) H q(a + b + y) H q(b + c + z + t) Hq(m)3 H q(a) 2 H q(b) H q(c) H q(x) H q(y) H q(z) H q(t) H q(m + a) 2 H q(m + b) H q(m + c) H q(x + t) H q(y + z), where = m + a + b + c.
31 Generalizing MacMahon s Formula c z+b x+y+d+e+f z+c b k O j i c z+b 7 x+y+d+e+f 6 z+c b x+a f e y+a x+a f e y+a 5 3 y+z+d+e+f d x+z+d+e+f y+z+d+e+f 8 4 d 2 x+z+d+e+f x+c a (a) y+b x+c a (b) y+b
32 Blum s Conjecture and Hexagonal Dungeon 2a=4 b=6 a=2 a=2 b=6 2a=4 Theorem (Blum s (ex-)conjecture) The number of tilings of the hexagonal dungeon of side-lengths a, 2a, b, a, 2a, b (b 2a) is 13 2a2 14 a2 2. The conjecture was proven by Ciucu and L. (2014).
33 Kuo condensation Theorem (Kuo)
Aztec diamond. An Aztec diamond of order n is the union of the unit squares with lattice point coordinates in the region given by...
Aztec diamond An Aztec diamond of order n is the union of the unit squares with lattice point coordinates in the region given by x + y n + 1 Aztec diamond An Aztec diamond of order n is the union of the
More informationPlane Tilings. Richard P. Stanley M.I.T. Plane Tilings p.
Plane Tilings p. Plane Tilings Richard P. Stanley M.I.T. Plane Tilings p. region: tiles: 1 2 3 4 5 6 7 Plane Tilings p. tiling: 4 3 7 5 6 2 1 Plane Tilings p. Is there a tiling? How many? About how many?
More informationTILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY
TILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY BERKELEY MATH CIRCLE 1. Looking for a number Consider an 8 8 checkerboard (like the one used to play chess) and consider 32
More informationarxiv: v4 [math.co] 7 Sep 2014
arxiv:1310.333v4 [math.co] 7 Sep 014 Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-octagonal regions TRI LAI Institute for Mathematics and Its Applications University of Minnesota Minneapolis,
More informationOn Domino Tilings of Rectangles
On Domino Tilings of Rectangles Aztec Diamonds in the Rough Trevor Bass trev@math.rutgers.edu (67) 699-962 Supervised by Professor Doron Zeilberger of Rutgers University. A thesis presented to the Department
More informationRandom Tilings with the GPU
Random Tilings with the GPU David Keating Joint work with A. Sridhar University of California, Berkeley June 8, 2018 1 / 33 Outline 1 2 3 4 Lozenge Tilings Six Vertex Bibone Tilings Rectangle-triangle
More informationINVERTING THE KASTELEYN MATRIX FOR SUB-GRAPHS OF THE HEXAGONAL LATTICE TOMACK GILMORE
INVERTING THE KASTELEYN MATRIX FOR SUB-GRAPHS OF THE HEXAGONAL LATTICE TOMACK GILMORE Fakultät für Mathematik der Universität Wien, Oskar-Morgernstern-Platz 1, 1090 Wien, Austria. Abstract. Consider a
More informationarxiv:math/ v2 [math.co] 11 Sep 2003
arxiv:math/0304090v2 [math.c] Sep 2003 Applications of Graphical Condensation for Enumerating Matchings and Tilings Eric H. Kuo August 22, 2003 Abstract A technique called graphical condensation is used
More informationLemma of Gessel Viennot
Advanced Graph Algorithms Jan-Apr 2014 Lemma of Gessel Viennot Speaker: Diptapriyo Majumdar Date: April 22, 2014 1 Overview In the last few lectures, we had learnt different algorithms using matrix multiplication.
More informationEnumeration Algorithm for Lattice Model
Enumeration Algorithm for Lattice Model Seungsang Oh Korea University International Workshop on Spatial Graphs 2016 Waseda University, August 5, 2016 Contents 1 State Matrix Recursion Algorithm 2 Monomer-Dimer
More informationSteep tilings and sequences of interlaced partitions
Steep tilings and sequences of interlaced partitions Jérémie Bouttier Joint work with Guillaume Chapuy and Sylvie Corteel arxiv:1407.0665 Institut de Physique Théorique, CEA Saclay Département de mathématiques
More informationAn Illustrative Study of the Enumeration of Tilings:
An Illustrative Study of the Enumeration of Tilings: Conjecture Discovery and Proof Techniques by Chris Douglas 1.0 Introduction We have exact formulas for the number of tilings of only a small number
More informationThe number of spanning trees of plane graphs with reflective symmetry
Journal of Combinatorial Theory, Series A 112 (2005) 105 116 www.elsevier.com/locate/jcta The number of spanning trees of plane graphs with reflective symmetry Mihai Ciucu a, Weigen Yan b, Fuji Zhang c
More informationEnumerating Tilings of Rectangles by Squares with Recurrence Relations
Journal of Combinatorics Volume 0, Number 0, 1, 2014 Enumerating Tilings of Rectangles by Squares with Recurrence Relations Daryl DeFord Counting the number of ways to tile an m n rectangle with squares
More informationNOTE. Rhombus Tilings of a Hexagon with Three Fixed Border Tiles
Journal of Combinatorial Theory, Series A 88, 368378 (1999) Article ID jcta.1999.3000, available online at http:www.idealibrary.com on Rhombus Tilings of a Hexagon with Three Fixed Border Tiles Theresia
More informationNegative Numbers in Combinatorics: Geometrical and Algebraic Perspectives
Negative Numbers in Combinatorics: Geometrical and Algebraic Perspectives James Propp (UMass Lowell) June 29, 2012 Slides for this talk are on-line at http://jamespropp.org/msri-up12.pdf 1 / 99 I. Equal
More informationBINARY CODES AND KASTELEYN 3-MATRICES MARTIN LOEBL AND PAVEL RYTÍŘ
BINARY CODES AND KASTELEYN -MATRICES MARTIN LOEBL AND PAVEL RYTÍŘ Department of Applied Mathematics, Charles University, Malostranské nám. 25, 8 00 Praha, Czech Republic. Abstract. Two cornerstones of
More informationMiquel dynamics for circle patterns
Miquel dynamics for circle patterns Sanjay Ramassamy ENS Lyon Partly joint work with Alexey Glutsyuk (ENS Lyon & HSE) Seminar of the Center for Advanced Studies Skoltech March 12 2018 Circle patterns form
More informationOrientations of Planar Graphs
Orientations of Planar Graphs Doc-Course Bellaterra March 11, 2009 Stefan Felsner Technische Universität Berlin felsner@math.tu-berlin.de Topics α-orientations Sample Applications Counting I: Bounds Counting
More informationACTUALLY DOING IT : an Introduction to Polyhedral Computation
ACTUALLY DOING IT : an Introduction to Polyhedral Computation Jesús A. De Loera Department of Mathematics Univ. of California, Davis http://www.math.ucdavis.edu/ deloera/ 1 What is a Convex Polytope? 2
More informationTrees and Matchings. James G. Propp. University of Wisconsin Madison, Wisconsin
Trees and Matchings arxiv:math/99325v2 [math.co] 3 Apr 2 Richard W. Kenyon Laboratoire de Topologie Université Paris-Sud kenyon@topo.math.u-psud.fr James G. Propp University of Wisconsin Madison, Wisconsin
More informationCovering the Aztec Diamond with One-sided Tetrasticks Extended Version
Covering the Aztec Diamond with One-sided Tetrasticks Extended Version Alfred Wassermann, University of Bayreuth, D-95440 Bayreuth, Germany Abstract There are 107 non-isomorphic coverings of the Aztec
More informationRandom Tilings. Thomas Fernique. Moscow, Spring 2011
Random Tilings Thomas Fernique Moscow, Spring 2011 1 Random tilings 2 The Dimer case 3 Random assembly 4 Random sampling 1 Random tilings 2 The Dimer case 3 Random assembly 4 Random sampling Quenching
More informationExact learning and inference for planar graphs
Supervisor: Nic Schraudolph September 2007 Ising model Particles modify their behaviour to conform with neighbours behaviour What is the mean energy? Used in chemistry, physics, biology... More than 12,000
More informationGraph Theoretical Cluster Expansion Formulas
Graph Theoretical Cluster Expansion Formulas Gregg Musiker MIT December 9, 008 http//math.mit.edu/ musiker/graphtalk.pdf Gregg Musiker (MIT) Graph Theoretical Cluster Expansions December 9, 008 / 9 Outline.
More informationThe Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs
The Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs Frankie Smith Nebraska Wesleyan University fsmith@nebrwesleyan.edu May 11, 2015 Abstract We will look at how to represent
More informationSampling Weighted Perfect Matchings on the Square-Octagon Lattice
Sampling Weighted Perfect Matchings on the Square-Octagon Lattice Prateek Bhakta Dana Randall Abstract We consider perfect matchings of the square-octagon lattice, also known as fortresses [16]. There
More informationDirectional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives
Recall that if z = f(x, y), then the partial derivatives f x and f y are defined as and represent the rates of change of z in the x- and y-directions, that is, in the directions of the unit vectors i and
More informationProper Partitions of a Polygon and k-catalan Numbers
Proper Partitions of a Polygon and k-catalan Numbers Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027 USA sagan@math.msu.edu July 13, 2005 Abstract Let P be
More informationZero-Sum Flow Numbers of Triangular Grids
Zero-Sum Flow Numbers of Triangular Grids Tao-Ming Wang 1,, Shih-Wei Hu 2, and Guang-Hui Zhang 3 1 Department of Applied Mathematics Tunghai University, Taichung, Taiwan, ROC 2 Institute of Information
More informationChapter 4. Triangular Sum Labeling
Chapter 4 Triangular Sum Labeling 32 Chapter 4. Triangular Sum Graphs 33 4.1 Introduction This chapter is focused on triangular sum labeling of graphs. As every graph is not a triangular sum graph it is
More informationLecture 9 - Matrix Multiplication Equivalences and Spectral Graph Theory 1
CME 305: Discrete Mathematics and Algorithms Instructor: Professor Aaron Sidford (sidford@stanfordedu) February 6, 2018 Lecture 9 - Matrix Multiplication Equivalences and Spectral Graph Theory 1 In the
More informationPower Set of a set and Relations
Power Set of a set and Relations 1 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}
More informationSPERNER S LEMMA MOOR XU
SPERNER S LEMMA MOOR XU Abstract. Is it possible to dissect a square into an odd number of triangles of equal area? This question was first answered by Paul Monsky in 970, and the solution requires elements
More informationDefinition. Given a (v,k,λ)- BIBD, (X,B), a set of disjoint blocks of B which partition X is called a parallel class.
Resolvable BIBDs Definition Given a (v,k,λ)- BIBD, (X,B), a set of disjoint blocks of B which partition X is called a parallel class. A partition of B into parallel classes (there must be r of them) is
More informationWeek 12: Trees; Review. 22 and 24 November, 2017
(1/24) MA284 : Discrete Mathematics Week 12: Trees; Review http://www.maths.nuigalway.ie/~niall/ma284/ 22 and 24 November, 2017 C C C C 1 Trees Recall... Applications: Chemistry Applications: Decision
More informationAlgebraic Graph Theory- Adjacency Matrix and Spectrum
Algebraic Graph Theory- Adjacency Matrix and Spectrum Michael Levet December 24, 2013 Introduction This tutorial will introduce the adjacency matrix, as well as spectral graph theory. For those familiar
More informationTILING RECTANGLES SIMON RUBINSTEIN-SALZEDO
TILING RECTANGLES SIMON RUBINSTEIN-SALZEDO. A classic tiling problem Question.. Suppose we tile a (large) rectangle with small rectangles, so that each small rectangle has at least one pair of sides with
More informationarxiv: v1 [math.gr] 19 Dec 2018
GROWTH SERIES OF CAT(0) CUBICAL COMPLEXES arxiv:1812.07755v1 [math.gr] 19 Dec 2018 BORIS OKUN AND RICHARD SCOTT Abstract. Let X be a CAT(0) cubical complex. The growth series of X at x is G x (t) = y V
More informationTessellations. Irena Swanson Reed College, Portland, Oregon. MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018
Tessellations Irena Swanson Reed College, Portland, Oregon MathPath, Lewis & Clark College, Portland, Oregon, 24 July 2018 What is a tessellation? A tiling or a tessellation of the plane is a covering
More informationStudent Research. UBC Math Circle, January 25, Richard Anstee UBC, Vancouver
Richard Anstee UBC, Vancouver UBC Math Circle, January 25, 2016 Dominoes and Matchings The first set of problems I d like to mention are really graph theory problems disguised as covering a checkerboard
More informationMatching and Planarity
Matching and Planarity Po-Shen Loh June 010 1 Warm-up 1. (Bondy 1.5.9.) There are n points in the plane such that every pair of points has distance 1. Show that there are at most n (unordered) pairs of
More information14.6 Directional Derivatives and the Gradient Vector
14 Partial Derivatives 14.6 and the Gradient Vector Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and the Gradient Vector In this section we introduce
More informationGraphs and Network Flows IE411. Lecture 21. Dr. Ted Ralphs
Graphs and Network Flows IE411 Lecture 21 Dr. Ted Ralphs IE411 Lecture 21 1 Combinatorial Optimization and Network Flows In general, most combinatorial optimization and integer programming problems are
More informationIntroduction III. Graphs. Motivations I. Introduction IV
Introduction I Graphs Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Graph theory was introduced in the 18th century by Leonhard Euler via the Königsberg
More informationIntroduction to. Graph Theory. Second Edition. Douglas B. West. University of Illinois Urbana. ftentice iiilil PRENTICE HALL
Introduction to Graph Theory Second Edition Douglas B. West University of Illinois Urbana ftentice iiilil PRENTICE HALL Upper Saddle River, NJ 07458 Contents Preface xi Chapter 1 Fundamental Concepts 1
More informationThe University of Sydney MATH 2009
The University of Sydney MTH 009 GRPH THORY Tutorial 10 Solutions 004 1. In a tournament, the score of a vertex is its out-degree, and the score sequence is a list of all the scores in non-decreasing order.
More informationWhich n-venn diagrams can be drawn with convex k-gons?
Which n-venn diagrams can be drawn with convex k-gons? Jeremy Carroll Frank Ruskey Mark Weston Abstract We establish a new lower bound for the number of sides required for the component curves of simple
More informationConway s Tiling Groups
Conway s Tiling Groups Elissa Ross Department of Mathematics University of British Columbia, BC, Canada elissa@math.ubc.ca December 12, 2004 Abstract In this paper I discuss a method of John Conway for
More informationPoints covered an odd number of times by translates
Points covered an odd number of times by translates Rom Pinchasi August 5, 0 Abstract Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of
More informationOn the Number of Tilings of a Square by Rectangles
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange University of Tennessee Honors Thesis Projects University of Tennessee Honors Program 5-2012 On the Number of Tilings
More informationOn the Relationships between Zero Forcing Numbers and Certain Graph Coverings
On the Relationships between Zero Forcing Numbers and Certain Graph Coverings Fatemeh Alinaghipour Taklimi, Shaun Fallat 1,, Karen Meagher 2 Department of Mathematics and Statistics, University of Regina,
More informationIntroductory Combinatorics
Introductory Combinatorics Third Edition KENNETH P. BOGART Dartmouth College,. " A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Toronto Sydney Tokyo xm CONTENTS
More informationEnumeration of Polyomino Tilings via Hypergraphs
Enumeration of Polyomino Tilings via Hypergraphs (Dedicated to Professor Károly Bezdek) Muhammad Ali Khan Centre for Computational and Discrete Geometry Department of Mathematics & Statistics, University
More informationDiscrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 2016-2017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.2. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations,
More informationHeffter Arrays: Biembeddings of Cycle Systems on Surfaces
Heffter Arrays: Biembeddings of Cycle Systems on Surfaces by Jeff Dinitz* University of Vermont and Dan Archdeacon (University of Vermont) Tom Boothby (Simon Fraser University) Our goal is to embed the
More informationKevin James. MTHSC 206 Section 15.6 Directional Derivatives and the Gra
MTHSC 206 Section 15.6 Directional Derivatives and the Gradient Vector Definition We define the directional derivative of the function f (x, y) at the point (x 0, y 0 ) in the direction of the unit vector
More informationLecture 5: Graphs. Rajat Mittal. IIT Kanpur
Lecture : Graphs Rajat Mittal IIT Kanpur Combinatorial graphs provide a natural way to model connections between different objects. They are very useful in depicting communication networks, social networks
More informationGRAPH THEORY and APPLICATIONS. Factorization Domination Indepence Clique
GRAPH THEORY and APPLICATIONS Factorization Domination Indepence Clique Factorization Factor A factor of a graph G is a spanning subgraph of G, not necessarily connected. G is the sum of factors G i, if:
More informationVertex Magic Total Labelings of Complete Graphs 1
Vertex Magic Total Labelings of Complete Graphs 1 Krishnappa. H. K. and Kishore Kothapalli and V. Ch. Venkaiah Centre for Security, Theory, and Algorithmic Research International Institute of Information
More informationr=1 The Binomial Theorem. 4 MA095/98G Revision
Revision Read through the whole course once Make summary sheets of important definitions and results, you can use the following pages as a start and fill in more yourself Do all assignments again Do the
More informationChapter 18 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal.
Chapter 8 out of 7 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 8 Matrices Definitions and Basic Operations Matrix algebra is also known
More informationDifferential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder]
Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder] Preliminaries Recall: Given a smooth function f:r R, the function
More informationPERFECT FOLDING OF THE PLANE
SOOCHOW JOURNAL OF MATHEMATICS Volume 32, No. 4, pp. 521-532, October 2006 PERFECT FOLDING OF THE PLANE BY E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN Abstract. In this paper we introduced the concept of
More informationPlanarity. 1 Introduction. 2 Topological Results
Planarity 1 Introduction A notion of drawing a graph in the plane has led to some of the most deep results in graph theory. Vaguely speaking by a drawing or embedding of a graph G in the plane we mean
More information#A14 INTEGERS 15A (2015) ON SUBSETS OF ORDERED TREES ENUMERATED BY A SUBSEQUENCE OF FIBONACCI NUMBERS
#A4 INTEGERS 5A (205) ON SUBSETS OF ORDERED TREES ENUMERATED BY A SUBSEQUENCE OF FIBONACCI NUMBERS Melkamu Zeleke Department of Mathematics, William Paterson University, Wayne, New Jersey Mahendra Jani
More informationREGULAR GRAPHS OF GIVEN GIRTH. Contents
REGULAR GRAPHS OF GIVEN GIRTH BROOKE ULLERY Contents 1. Introduction This paper gives an introduction to the area of graph theory dealing with properties of regular graphs of given girth. A large portion
More informationExercise set 2 Solutions
Exercise set 2 Solutions Let H and H be the two components of T e and let F E(T ) consist of the edges of T with one endpoint in V (H), the other in V (H ) Since T is connected, F Furthermore, since T
More informationarxiv: v2 [math.co] 26 Jul 2016
Generalizing the divisibility property of rectangle domino tilings arxiv:1406.0669v2 [math.co] 26 Jul 2016 Forest Tong Department of Mathematics Massachusetts Institute of Technology Cambridge, MA, USA
More informationCourse Introduction / Review of Fundamentals of Graph Theory
Course Introduction / Review of Fundamentals of Graph Theory Hiroki Sayama sayama@binghamton.edu Rise of Network Science (From Barabasi 2010) 2 Network models Many discrete parts involved Classic mean-field
More informationPlanar Drawing of Bipartite Graph by Eliminating Minimum Number of Edges
UITS Journal Volume: Issue: 2 ISSN: 2226-32 ISSN: 2226-328 Planar Drawing of Bipartite Graph by Eliminating Minimum Number of Edges Muhammad Golam Kibria Muhammad Oarisul Hasan Rifat 2 Md. Shakil Ahamed
More informationarxiv: v1 [math.co] 20 Aug 2012
ENUMERATING TRIANGULATIONS BY PARALLEL DIAGONALS Alon Regev Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois regev@math.niu.edu arxiv:108.91v1 [math.co] 0 Aug 01 1 Introduction
More informationClick the mouse button or press the Space Bar to display the answers.
Click the mouse button or press the Space Bar to display the answers. 9-4 Objectives You will learn to: Identify regular tessellations. Vocabulary Tessellation Regular Tessellation Uniform Semi-Regular
More informationSemistandard Young Tableaux Polytopes. Sara Solhjem Joint work with Jessica Striker. April 9, 2017
Semistandard Young Tableaux Polytopes Sara Solhjem Joint work with Jessica Striker North Dakota State University Graduate Student Combinatorics Conference 217 April 9, 217 Sara Solhjem (NDSU) Semistandard
More informationOn FPL configurations with four sets of nested arches
SPhT-T04/030 arxiv:cond-mat/0403268v2 [cond-mat.stat-mech] 18 Mar 2004 On FPL configurations with four sets of nested arches P. Di Francesco and J.-B. Zuber Service de Physique Théorique de Saclay, CEA/DSM/SPhT,
More informationTiling with fences. 16:00 Wednesday 26th October 2016, Room 3303, MUIC. Michael A. Allen. Physics Department, Mahidol University, Bangkok
Tiling with fences 16:00 Wednesday 26th October 2016, Room 3303, MUIC Michael A. Allen Physics Department, Mahidol University, Bangkok We look at tiling an n-board (a linear array of n square cells of
More informationApollonian ring packings
Apollonian ring packings Dedicated to Ron Graham Adrian Bolt Steve Butler Espen Hovland Abstract Apollonian packings are a well-known object in mathematics formed by starting with a set of three mutually
More informationCounting the number of spanning tree. Pied Piper Department of Computer Science and Engineering Shanghai Jiao Tong University
Counting the number of spanning tree Pied Piper Department of Computer Science and Engineering Shanghai Jiao Tong University 目录 Contents 1 Complete Graph 2 Proof of the Lemma 3 Arbitrary Graph 4 Proof
More informationOctonion multiplication and Heawood s map
Octonion multiplication and Heawood s map Bruno Sévennec arxiv:0.0v [math.ra] 29 Jun 20 June 30, 20 Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly)
More informationCMSC Honors Discrete Mathematics
CMSC 27130 Honors Discrete Mathematics Lectures by Alexander Razborov Notes by Justin Lubin The University of Chicago, Autumn 2017 1 Contents I Number Theory 4 1 The Euclidean Algorithm 4 2 Mathematical
More informationPlanar Graphs with Many Perfect Matchings and Forests
Planar Graphs with Many Perfect Matchings and Forests Michael Biro Abstract We determine the number of perfect matchings and forests in a family T r,3 of triangulated prism graphs. These results show that
More informationWinning Positions in Simplicial Nim
Winning Positions in Simplicial Nim David Horrocks Department of Mathematics and Statistics University of Prince Edward Island Charlottetown, Prince Edward Island, Canada, C1A 4P3 dhorrocks@upei.ca Submitted:
More informationplanar graph embeddings and stat mech
planar graph embeddings and stat mech Richard Kenyon (Brown University) In 2D stat mech models, appropriate graph embeddings are important e.g. Bond percolation on Z 2. p c = 1 2 What about unequal probabilities?
More informationMapping Common Core State Standard Clusters and. Ohio Grade Level Indicator. Grade 5 Mathematics
Mapping Common Core State Clusters and Ohio s Grade Level Indicators: Grade 5 Mathematics Operations and Algebraic Thinking: Write and interpret numerical expressions. Operations and Algebraic Thinking:
More informationAcyclic Subgraphs of Planar Digraphs
Acyclic Subgraphs of Planar Digraphs Noah Golowich Research Science Institute Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts, U.S.A. ngolowich@college.harvard.edu
More informationVertex Magic Total Labelings of Complete Graphs
AKCE J. Graphs. Combin., 6, No. 1 (2009), pp. 143-154 Vertex Magic Total Labelings of Complete Graphs H. K. Krishnappa, Kishore Kothapalli and V. Ch. Venkaiah Center for Security, Theory, and Algorithmic
More informationRandom strongly regular graphs?
Graphs with 3 vertices Random strongly regular graphs? Peter J Cameron School of Mathematical Sciences Queen Mary, University of London London E1 NS, U.K. p.j.cameron@qmul.ac.uk COMB01, Barcelona, 1 September
More informationTransformations. Write three rules based on what you figured out above: To reflect across the y-axis. (x,y) To reflect across y=x.
Transformations Geometry 14.1 A transformation is a change in coordinates plotted on the plane. We will learn about four types of transformations on the plane: Translations, Reflections, Rotations, and
More informationEnumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry
Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry Hiroshi Fukuda 1, Nobuaki Mutoh 2, Gisaku Nakamura 3, and Doris Schattschneider 4 1 College of Liberal
More information2. Draw a non-isosceles triangle. Now make a template of this triangle out of cardstock or cardboard.
Tessellations The figure at the left shows a tiled floor. Because the floor is entirely covered by the tiles we call this arrangement a tessellation of the plane. A regular tessellation occurs when: The
More informationTangencies between disjoint regions in the plane
June 16, 20 Problem Definition Two nonoverlapping Jordan regions in the plane are said to touch each other or to be tangent to each other if their boundaries have precisely one point in common and their
More informationCOUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.3 Counting Elements of Disjoint Sets: The Addition Rule Copyright Cengage Learning. All rights reserved. Counting
More informationResolutions of the pair design, or 1-factorisations of complete graphs. 1 Introduction. 2 Further constructions
Resolutions of the pair design, or 1-factorisations of complete graphs 1 Introduction A resolution of a block design is a partition of the blocks of the design into parallel classes, each of which forms
More informationLecture 3A: Generalized associahedra
Lecture 3A: Generalized associahedra Nathan Reading NC State University Cluster Algebras and Cluster Combinatorics MSRI Summer Graduate Workshop, August 2011 Introduction Associahedron and cyclohedron
More information2017 SOLUTIONS (PRELIMINARY VERSION)
SIMON MARAIS MATHEMATICS COMPETITION 07 SOLUTIONS (PRELIMINARY VERSION) This document will be updated to include alternative solutions provided by contestants, after the competition has been mared. Problem
More informationLozenge Tiling Constrained Codes
SUBMITTED TO IEE TRANSACTIONS ON COMMUNICATION 1 Lozenge Tiling Constrained Codes Bane Vasić, Fellow, IEEE and Anantha Raman Krishnan, Member, IEEE Abstract While the field of one-dimensional constrained
More informationThe extendability of matchings in strongly regular graphs
The extendability of matchings in strongly regular graphs Sebastian Cioabă Department of Mathematical Sciences University of Delaware Villanova, June 5, 2014 Introduction Matching A set of edges M of a
More informationOn ɛ-unit distance graphs
On ɛ-unit distance graphs Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 g-exoo@indstate.edu April 9, 003 Abstract We consider a variation on
More informationarxiv: v1 [math.gr] 2 Oct 2013
POLYGONAL VH COMPLEXES JASON K.C. POLÁK AND DANIEL T. WISE arxiv:1310.0843v1 [math.gr] 2 Oct 2013 Abstract. Ian Leary inquires whether a class of hyperbolic finitely presented groups are residually finite.
More informationMATH 350 GRAPH THEORY & COMBINATORICS. Contents
MATH 350 GRAPH THEORY & COMBINATORICS PROF. SERGEY NORIN, FALL 2013 Contents 1. Basic definitions 1 2. Connectivity 2 3. Trees 3 4. Spanning Trees 3 5. Shortest paths 4 6. Eulerian & Hamiltonian cycles
More information