Enumerative Combinatoric Algorithms. Pólya-Redfield Enumeration Theorem aka Burnside s Lemma
|
|
|
- Christian Cooper
- 6 years ago
- Views:
Transcription
1 Enumerative Combinatoric Algorithms Pólya-Redfield Enumeration Theorem aka Burnside s Lemma J.H. Redfield (1927), G. Pólya (1937): combinatorics W. Burnside (1911): orbit based Oswin Aichholzer: Enumerative Combinatoric Algorithms,
2 Some Examples for Starters 1 A) Consider strings of length 4 with characters {A, B, C}. Operation: cyclic shift by i positions, 0 i 3. How many different strings exist? B) Consider a 3 3 grid as shown. Color the unit squares with 3 different colors. How many different colorings exist, if we allow (a) rotation (b) rotation and reflection? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
3 Some Examples for Starters 2 C) For a 3D cube color the 6 faces with k colors. How many different colorings are possible, if you can rotate the cube as usual in 3-space? D) How many different positions in Tic Tac Toe can we get after k half-moves, considering symmetries. Oswin Aichholzer: Enumerative Combinatoric Algorithms,
4 Objects and Operations Set of objects X Set of n operations R = {R i, 0 i n 1} R forms a group [1] w.r.t. (4 axioms R i, R j ): Closure: R k : R i R j = R k Associativity: (R i R j ) R k = R i (R j R k ) R 0 is identity element, i.e., R 0 R i = R i R 0 = R i Inverse element: R k : R k R i = R i R k = R 0 Objects might be strings, colored grids, colored cubes,... Operations might be rotations, reflections,... For objects R 0 is the identity (neutral) function. [1]: does not need to be commutative: it might well be that R i R j R j R i Oswin Aichholzer: Enumerative Combinatoric Algorithms,
5 Orbits An orbit is the set of all objects from X which can be transformed into each other by an operation from R (equivalence class). The length of an orbit is the number of elements it contains. Main Question: How many orbits exist? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
6 Stabilizers For an object x X the operation R i R is a stabilizer if and only if R i (x) = x. m x... number of stabilizers for x r i... invariance number, i.e., number of objects for which R i is a stabilizer. We have n 1 i=0 r i = x X m x by double counting (each pair operation/object which gives an invariance is counted once on each side; left side counts by operation, right side counts by objects). Oswin Aichholzer: Enumerative Combinatoric Algorithms,
7 Counting Orbits 1 Deriving a first relation simplified version (not valid in full generality): x orbit m x = n This implies is unchanged! x X m x = Iterate the process. all orbits x orbit m x = all orbits n = n times the number of orbits. Thus one orbit n multiobjects All objects different: n times identity as stabilizer. Two times the same object: unify the vertices! For removed copy: remove outgoin edges, reroute incoming edges. We loose on identity as stabilizer, but get one new stabilizer, so the total number number of orbits = x X m x n Oswin Aichholzer: Enumerative Combinatoric Algorithms,
8 Counting Orbits 2 And with the previous observation n 1 i=0 r i = x X m x we therefore get number of orbits = n 1 i=0 r i n We will use this equation, as usually n << X and thus, all invariance numbers r i are easier to obtain than all numbers m x of stabilizers. Oswin Aichholzer: Enumerative Combinatoric Algorithms,
9 Examples: Strings (1) Consider strings of length 4 with characters {A, B}. Operation: cyclic shift by i positions, 0 i 3. How many different strings exist? Algorithm: 1. Identify all operations R 0 to R n 1 2. Compute all invariance numbers r i, 0 i n 1 3. Compute the number of orbits by n 1 i=0 r i n Oswin Aichholzer: Enumerative Combinatoric Algorithms,
10 Examples: Strings (2) Consider strings of length 4 with characters {A, B, C}. Operation: cyclic shift by i positions, 0 i 3. How many different strings exist? [Example A)] Oswin Aichholzer: Enumerative Combinatoric Algorithms,
11 Examples: Strings (3) Consider strings of length 4 with characters {A, B, C}, where each character is used at least once. Operation: cyclic shift by i positions, 0 i 3. How many different strings exist? (cf. Chapter about Inclusion-Exclusion Principle) Oswin Aichholzer: Enumerative Combinatoric Algorithms,
12 Examples: Grids (4) Consider a 2 2 grid as shown. Color the unit squares with 2 different colors. How many different colorings exist, if we allow (a) rotation (b) rotation and reflection? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
13 Examples: Grids (5) Consider a 2 2 grid as shown. Color the unit squares with 3 different colors. How many different colorings exist, if we allow (a) rotation (b) rotation and reflection (c) only reflection? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
14 Examples: Grids (6) Consider a 3 3 grid as shown. Color the unit squares with 2 different colors. How many different colorings exist, if we allow (a) rotation (b) rotation and reflection? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
15 Examples: Grids (7) Consider a 3 3 grid as shown. Color the unit squares with 3 different colors. How many different colorings exist, if we allow (a) rotation (b) rotation and reflection? [Example B)] Oswin Aichholzer: Enumerative Combinatoric Algorithms,
16 Examples: Cube (8) For a 3D cube color the 6 faces with 2 colors. How many different colorings are possible, if you can rotate the cube as usual in 3-space? Oswin Aichholzer: Enumerative Combinatoric Algorithms,
17 Examples: Cube (9) For a 3D cube color the 6 faces with 3 and, more general, with k colors. How many different colorings are possible, if you can rotate the cube as usual in 3-space? [Example C)] Oswin Aichholzer: Enumerative Combinatoric Algorithms,
18 Examples: Cube (10) For a 3D cube place the numbers 1 to 6 on the 6 faces. How many different labelings are possible, if you can rotate the cube as usual in 3-space? The orientation of a number within a face can be ignored, that is, we only consider the assignement of numbers to faces. Oswin Aichholzer: Enumerative Combinatoric Algorithms,
19 Examples: Tik Tak Toe (11) Enumerate all different valid game positions of Tic Tac Toe after k half-moves, considering symmetries. o starts, x follows k+1 2 o, and k 2 x tokens are placed after k half-moves. [Example D)] Oswin Aichholzer: Enumerative Combinatoric Algorithms,
Discrete Mathematics lecture notes 13-2 December 8, 2013
Discrete Mathematics lecture notes 13-2 December 8, 2013 34. Pólya s Enumeration Theorem We started examining the number of essentially distinct colorings of an object with questions of the form How many
Enumerative Combinatorics 7: Group actions
Enumerative Combinatorics 7: Group actions Peter J. Cameron Autumn 2013 How many ways can you colour the faces of a cube with three colours? Clearly the answer is 3 6 = 729. But what if we regard two colourings
ON THE NUMBER OF HAMILTONIAN CIRCUITS IN THE «-CUBE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, July 1975 ON THE NUMBER OF HAMILTONIAN CIRCUITS IN THE «-CUBE E. DIXON AND S. GOODMAN ABSTRACT. Improved upper and lower bounds are found for
Homework #4 Solutions Due: July 3, Do the following exercises from Lax: Page 124: 9.1, 9.3, 9.5
Do the following exercises from Lax: Page 124: 9.1, 9.3, 9.5 9.1. a) Find the number of different squares with vertices colored red, white, or blue. b) Find the number of different m-colored squares for
10 Steps to Counting Unlabeled Planar Graphs: 20 Years Later
10 Steps to : 20 Years Later Manuel Bodirsky October 2007 A005470 Sloane Sequence A005470 (core, nice, hard): Number p(n) of unlabeled planar simple graphs with n nodes. Initial terms: 1, 2, 4, 11, 33,
PÓLYA S COUNTING THEORY Mollee Huisinga May 9, 2012
PÓLYA S COUNTING THEORY Mollee Huisinga May 9, 202 Introduction In combinatorics, there are very few formulas that apply comprehensively to all cases of a given problem. Pólya s Counting Theory is a spectacular
An Introduction to Combinatorics
Chapter 1 An Introduction to Combinatorics What Is Combinatorics? Combinatorics is the study of how to count things Have you ever counted the number of games teams would play if each team played every
POLYA S ENUMERATION ALEC ZHANG
POLYA S ENUMERATION ALEC ZHANG Abstract. We explore Polya s theory of counting from first principles, first building up the necessary algebra and group theory before proving Polya s Enumeration Theorem
How to count - an exposition of Polya s theory of enumeration
How to count - an exposition of Polya s theory of enumeration Shriya Anand Published in Resonance, September 2002 P.19-35. Shriya Anand is a BA Honours Mathematics III year student from St. Stephens College,
What is a semigroup? What is a group? What is the difference between a semigroup and a group?
The second exam will be on Thursday, July 5, 2012. The syllabus will be Sections IV.5 (RSA Encryption), III.1, III.2, III.3, III.4 and III.8, III.9, plus the handout on Burnside coloring arguments. Of
Enumerating Distinct Chessboard Tilings and Generalized Lucas Sequences Part I
Enumerating Distinct Chessboard Tilings and Generalized Lucas Sequences Part I Daryl DeFord Washington State University January 28, 2013 Roadmap 1 Problem 2 Symmetry 3 LHCCRR as Vector Spaces 4 Generalized
Chapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
SYMMETRIES IN R 3 NAMITA GUPTA
SYMMETRIES IN R 3 NAMITA GUPTA Abstract. This paper will introduce the concept of symmetries being represented as permutations and will proceed to explain the group structure of such symmetries under composition.
Notes on cycle generating functions
Notes on cycle generating functions. The cycle generating function of a species A combinatorial species is a rule attaching to each finite set X a set of structures on X, in such a way that the allowed
BURNSIDE S LEMMA. 1. Discriminating Dice Using 3 Colors
International Journal of Pure and Applied Mathematics Volume 82 No. 3 2013, 431-440 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu PA ijpam.eu BURNSIDE S LEMMA Yutaka Nishiyama Department of
18. Counting Patterns
18.1 The Problem of Counting Patterns 18. Counting Patterns For this discussion, consider a collection of objects and a group of permutation symmetries (G) that can act on the objects. An object is not
KLEIN LINK MULTIPLICITY AND RECURSION. 1. Introduction
KLEIN LINK MULTIPLICITY AND RECURSION JENNIFER BOWEN, DAVID FREUND, JOHN RAMSAY, AND SARAH SMITH-POLDERMAN Abstract. The (m, n)-klein links are formed by altering the rectangular representation of an (m,
Enumeration and symmetry of edit metric spaces. Jessie Katherine Campbell. A dissertation submitted to the graduate faculty
Enumeration and symmetry of edit metric spaces by Jessie Katherine Campbell A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY
Name: Solutions - AI FINAL EXAM
1 2 3 4 5 6 7 8 9 10 11 12 13 total Name: Solutions - AI FINAL EXAM The first 7 problems will each count 10 points. The best 3 of # 8-13 will count 10 points each. Total is 100 points. A 4th problem from
Necklaces Pólya s Enumeration Theorem. Pólya Counting I. Gordon Royle. Semester 1, 2004
Semester 1, 2004 George Pólya (1887 1985) George Polya discovered a powerful general method for enumerating the number of orbits of a group on particular configurations. This method became known as the
Review problems solutions
Review problems solutions Math 3152 December 15, 2017 1. Use the binomial theorem to prove that, for all n 1 and k satisfying 0 k n, ( )( ) { n i ( 1) i k 1 if k n; i k 0 otherwise. ik Solution: Using
k k k 1 Lecture 9: Applying Backpropagation Lecture 9: Applying Backpropagation 3 Lecture 9: Applying Backpropagation
K-Class Classification Problem Let us denote the -th class by C, with n exemplars or training samples, forming the sets T for = 1,, K: {( x, ) p = 1 n } T = d,..., p p The complete training set is T =
MATRIX INTEGRALS AND MAP ENUMERATION 2
MATRIX ITEGRALS AD MAP EUMERATIO 2 IVA CORWI Abstract. We prove the generating function formula for one face maps and for plane diagrams using techniques from Random Matrix Theory and orthogonal polynomials.
Polya theory (With a radar application motivation)
Polya theory (With a radar application motivation) Daniel Zwillinger, PhD 5 April 200 Sudbury --623 telephone 4 660 Daniel I Zwillinger@raytheon.com http://www.az-tec.com/zwillinger/talks/2000405/ http://nesystemsengineering.rsc.ray.com/training/memo.polya.pdf
(a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict.)
![(a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict.)](/thumbs/89/99851024.jpg "(a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict.)")
1 Enumeration 11 Basic counting principles 1 June 2008, Question 1: (a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict) n/2 ( ) n (b) Find a closed form
Groups in Cryptography. Çetin Kaya Koç Winter / 13
http://koclab.org Çetin Kaya Koç Winter 2017 1 / 13 A set S and a binary operation A group G = (S, ) if S and satisfy: Closure: If a, b S then a b S Associativity: For a, b, c S, (a b) c = a (b c) A neutral
Counting. Mukulika Ghosh. Fall Based on slides by Dr. Hyunyoung Lee
Counting Mukulika Ghosh Fall 2018 Based on slides by Dr. Hyunyoung Lee Counting Counting The art of counting is known as enumerative combinatorics. One tries to count the number of elements in a set (or,
Materials and Handouts - Warm-Up - Answers to homework #1 - Keynote and notes template - Tic Tac Toe grids - Homework #2
Calculus Unit 1, Lesson 2: Composite Functions DATE: Objectives The students will be able to: - Evaluate composite functions using all representations Simplify composite functions Materials and Handouts
Notes on counting. James Aspnes. December 13, 2010
Notes on counting James Aspnes December 13, 2010 1 What counting is Recall that in set theory we formally defined each natural number as the set of all smaller natural numbers, so that n = {0, 1, 2,...,
Computation of the cycle index polynomial of a Permutation Group CS497-report
Computation of the cycle index polynomial of a Permutation Group CS497-report Rohit Gurjar Y5383 Supervisor: Prof Piyush P. Kurur Dept. Of Computer Science and Engineering, IIT Kanpur November 3, 2008
Course Info. Instructor: Office hour: 804, Tuesday, 2pm-4pm course homepage:
Combinatorics Course Info Instructor: yinyt@nju.edu.cn, yitong.yin@gmail.com Office hour: 804, Tuesday, 2pm-4pm course homepage: http://tcs.nju.edu.cn/wiki/ Textbook van Lint and Wilson, A course in Combinatorics,
A Course in Combinatorics
A Course in Combinatorics J. H. van Lint Technical Universüy of Eindhoven and R. M. Wilson California Institute of Technology H CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface xi 1. Graphs 1 Terminology of
Lecture 4: Orbits. Rajat Mittal. IIT Kanpur
Lecture 4: Orbits Rajat Mittal IIT Kanpur In the beginning of the course we asked a question. How many different necklaces can we form using 2 black beads and 10 white beads? In the question, the numbers
Unavoidable patterns in words
Unavoidable patterns in words Benny Sudakov ETH, Zurich joint with D.Conlon and J. Fox Ramsey numbers Definition: The Ramsey number r k (n) is the minimum N such that every 2-coloring of the k-tuples of
Algebra: Groups. Group Theory a. Examples of Groups. groups. The inverse of a is simply a, which exists.
Group Theory a Let G be a set and be a binary operation on G. (G, ) is called a group if it satisfies the following. 1. For all a, b G, a b G (closure). 2. For all a, b, c G, a (b c) = (a b) c (associativity).
Combining the cycle index and the Tutte polynomial?
Combining the cycle index and the Tutte polynomial? Peter J. Cameron University of St Andrews Combinatorics Seminar University of Vienna 23 March 2017 Selections Students often meet the following table
Notation. Irrep labels follow Mulliken s convention: A and B label nondegenerate E irreps, doubly T degenerate, triply degenerate
Notation Irrep labels follow Mulliken s convention: A and B label nondegenerate E irreps, doubly T degenerate, triply degenerate Spectroscopists sometimes use F for triply degenerate; almost everyone G
L(x; 0) = f(x) This family is equivalent to that obtained by the heat or diffusion equation: L t = 1 2 L
Figure 1: Fine To Coarse Scales. The parameter t is often associated with the generation of a scale-space family, where t = 0 denotes the original image and t > 0 increasing (coarser) scales. Scale-Space
Recursive Definitions
Recursive Definitions Example: Give a recursive definition of a n. a R and n N. Basis: n = 0, a 0 = 1. Recursion: a n+1 = a a n. Example: Give a recursive definition of n i=0 a i. Let S n = n i=0 a i,
Skew-closed objects, typings of linear lambda terms, and flows on trivalent graphs
Skew-closed objects, typings of linear lambda terms, and flows on trivalent graphs Noam Zeilberger University of Birmingham STRING 2017, Jericho Tavern 10 September 2017 [work in progress, in part based
Replica Condensation and Tree Decay
Replica Condensation and Tree Decay Arthur Jaffe and David Moser Harvard University Cambridge, MA 02138, USA Arthur Jaffe@harvard.edu, David.Moser@gmx.net June 7, 2007 Abstract We give an intuitive method
Counting Colorings Cleverly
Counting Colorings Cleverly by Zev Chonoles How many ways are there to color a shape? Of course, the answer depends on the number of colors we re allowed to use. More fundamentally, the answer depends
The Inclusion Exclusion Principle and Its More General Version
The Inclusion Exclusion Principle and Its More General Version Stewart Weiss June 28, 2009 1 Introduction The Inclusion-Exclusion Principle is typically seen in the context of combinatorics or probability
Discrete Mathematics. Benny George K. September 22, 2011
Discrete Mathematics Benny George K Department of Computer Science and Engineering Indian Institute of Technology Guwahati ben@iitg.ernet.in September 22, 2011 Set Theory Elementary Concepts Let A and
q xk y n k. , provided k < n. (This does not hold for k n.) Give a combinatorial proof of this recurrence by means of a bijective transformation.
Math 880 Alternative Challenge Problems Fall 2016 A1. Given, n 1, show that: m1 m 2 m = ( ) n+ 1 2 1, where the sum ranges over all positive integer solutions (m 1,..., m ) of m 1 + + m = n. Give both
CW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
BLOCK 4 ~ SYSTEMS OF EQUATIONS
BLOCK 4 ~ SYSTEMS OF EQUATIONS TIC-TAC-TOE How Many Solutions? Pros and Cons Polygons Determine whether a system of three linear equations has zero, one or infinitely many solutions. Create a visual display
Topological dynamics: basic notions and examples
CHAPTER 9 Topological dynamics: basic notions and examples We introduce the notion of a dynamical system, over a given semigroup S. This is a (compact Hausdorff) topological space on which the semigroup
On cycle index and orbit stabilizer of Symmetric group
The International Journal Of Engineering And Science (IJES) Volume 3 Issue 1 Pages 18-26 2014 ISSN (e): 2319 1813 ISSN (p): 2319 1805 On cycle index and orbit stabilizer of Symmetric group 1 Mogbonju M.
INEQUALITIES OF SYMMETRIC FUNCTIONS. 1. Introduction to Symmetric Functions [?] Definition 1.1. A symmetric function in n variables is a function, f,
![INEQUALITIES OF SYMMETRIC FUNCTIONS. 1. Introduction to Symmetric Functions [?] Definition 1.1. A symmetric function in n variables is a function, f,](/thumbs/87/97372587.jpg "INEQUALITIES OF SYMMETRIC FUNCTIONS. 1. Introduction to Symmetric Functions [?] Definition 1.1. A symmetric function in n variables is a function, f,")
INEQUALITIES OF SMMETRIC FUNCTIONS JONATHAN D. LIMA Abstract. We prove several symmetric function inequalities and conjecture a partially proved comprehensive theorem. We also introduce the condition of
Math 120: Homework 2 Solutions
Math 120: Homework 2 Solutions October 12, 2018 Problem 1.2 # 9. Let G be the group of rigid motions of the tetrahedron. Show that G = 12. Solution. Let us label the vertices of the tetrahedron 1, 2, 3,
The generating function of simultaneous s/t-cores
The generating function of simultaneous s/t-cores William J. Keith, wjk150@cii.fc.ul.pt University of Lisbon CMUC Seminar, Univ. Coimbra Abstract Previous work on partitions simultaneously s-core and t-core
Lecture 2: Mutually Orthogonal Latin Squares and Finite Fields
Latin Squares Instructor: Padraic Bartlett Lecture 2: Mutually Orthogonal Latin Squares and Finite Fields Week 2 Mathcamp 2012 Before we start this lecture, try solving the following problem: Question
Fundamentals of Coding For Network Coding
Fundamentals of Coding For Network Coding Marcus Greferath and Angeles Vazquez-Castro Aalto University School of Sciences, Finland Autonomous University of Barcelona, Spain marcus.greferath@aalto.fi and
1 st Grade LEUSD Learning Targets in Mathematics
1 st Grade LEUSD Learning Targets in Mathematics 8/21/2015 The learning targets below are intended to provide a guide for teachers in determining whether students are exhibiting characteristics of being
The cycle polynomial of a permutation group
The cycle polynomial of a permutation group Peter J. Cameron School of Mathematics and Statistics University of St Andrews North Haugh St Andrews, Fife, U.K. pjc0@st-andrews.ac.uk Jason Semeraro Department
Math 463/563 Homework #2 - Solutions
Math 463/563 Homework #2 - Solutions 1. How many 5-digit numbers can be formed with the integers 1, 2,..., 9 if no digit can appear exactly once? (For instance 43443 is OK, while 43413 is not allowed.)
The Strong Interaction and LHC phenomenology
The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules
Enumeration of Chord Diagrams without Loops and Parallel Chords
Enumeration of Chord Diagrams without Loops and Parallel Chords Evgeniy Krasko Alexander Omelchenko St. Petersburg Academic University 8/3 Khlopina Street, St. Petersburg, 9402, Russia {krasko.evgeniy,
Free fields, Quivers and Riemann surfaces
Free fields, Quivers and Riemann surfaces Sanjaye Ramgoolam Queen Mary, University of London 11 September 2013 Quivers as Calculators : Counting, correlators and Riemann surfaces, arxiv:1301.1980, J. Pasukonis,
On representable graphs
On representable graphs Sergey Kitaev and Artem Pyatkin 3rd November 2005 Abstract A graph G = (V, E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in
May 2 nd May 6 th. Unit 10: Rational & Irrational Numbers
Math 8: Week 33 Math Packet May 2 nd May 6 th Unit 10: Rational & Irrational Numbers Jump Start Directions: Fill out all the BINGO boards on pages 1-2 with the following perfect square numbers: 2, 4, 9,
ENUMERATION OF CONNECTED CATALAN OBJECTS BY TYPE. 1. Introduction
ENUMERATION OF CONNECTED CATALAN OBJECTS BY TYPE BRENDON RHOADES Abstract. Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects
2030 LECTURES. R. Craigen. Inclusion/Exclusion and Relations
2030 LECTURES R. Craigen Inclusion/Exclusion and Relations The Principle of Inclusion-Exclusion 7 ROS enumerates the union of disjoint sets. What if sets overlap? Some 17 out of 30 students in a class
A Mathematical Analysis of The Generalized Oval Track Puzzle
Rose-Hulman Undergraduate Mathematics Journal Volume 12 Issue 1 Article 5 A Mathematical Analysis of The Generalized Oval Track Puzzle Samuel Kaufmann Carnegie Mellon University, sakaufma@andrew.cmu.edu
are the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1).
Lecture A jacques@ucsd.edu Notation: N, R, Z, F, C naturals, reals, integers, a field, complex numbers. p(n), S n,, b(n), s n, partition numbers, Stirling of the second ind, Bell numbers, Stirling of the
Properties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
On Vector Product Algebras
On Vector Product Algebras By Markus Rost. This text contains some remarks on vector product algebras and the graphical techniques. It is partially contained in the diploma thesis of D. Boos and S. Maurer.
a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
How to Count- An Exposition of Polya's Theory of Enumeration
I ARTICLE How to Count- An Exposition of Polya's Theory of Enumeration Shriya Anand Consider the following problem: Mr.A has a cube whose faces he wants to paint either red or green. He wants to know how
Internet Monetization
Internet Monetization March May, 2013 Discrete time Finite A decision process (MDP) is reward process with decisions. It models an environment in which all states are and time is divided into stages. Definition
Huntington Beach City School District Grade 1 Mathematics Standards Schedule
2016-2017 Interim Assessment Schedule Orange Interim Assessment: November 1 - November 18, 2016 Green Interim Assessment: February 20 - March 10, 2017 Blueprint Summative Assessment: May 29 - June 16,
WHAT IS a sandpile? Lionel Levine and James Propp
WHAT IS a sandpile? Lionel Levine and James Propp An abelian sandpile is a collection of indistinguishable chips distributed among the vertices of a graph. More precisely, it is a function from the vertices
Angular Momentum Algebra
Angular Momentum Algebra Chris Clark August 1, 2006 1 Input We will be going through the derivation of the angular momentum operator algebra. The only inputs to this mathematical formalism are the basic
Combinatorics and Graph Theory NISER-MA401 Semester 4 of 2010
Combinatorics and Graph Theory NISER-MA401 Semester 4 of 2010 Instructor: Deepak Kumar Dalai 6th January 2010 1. Elementary Enumeration [3, Chapter 1] (a Some basic counting i. Counting the number of strings
Lecture 7 Feb 4, 14. Sections 1.7 and 1.8 Some problems from Sec 1.8
Lecture 7 Feb 4, 14 Sections 1.7 and 1.8 Some problems from Sec 1.8 Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Nonexistence Proofs Uniqueness
Unlabeled Motzkin numbers
Dept. Computer Science and Engineering 013 Catalan numbers Catalan numbers can be defined by the explicit formula: C n = 1 n = n! n + 1 n n!n + 1! and the ordinary generating function: Cx = n=0 C n x n
RAMSEY THEORY. 1 Ramsey Numbers
RAMSEY THEORY 1 Ramsey Numbers Party Problem: Find the minimum number R(k, l) of guests that must be invited so that at least k will know each other or at least l will not know each other (we assume that
CHAPTER 2 -idempotent matrices
CHAPTER 2 -idempotent matrices A -idempotent matrix is defined and some of its basic characterizations are derived (see [33]) in this chapter. It is shown that if is a -idempotent matrix then it is quadripotent
MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION
MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION SPRING 2014 - MOON Write your answer neatly and show steps. Any electronic devices including calculators, cell phones are not allowed. (1) Write the definition.
B7 Symmetry : Questions
B7 Symmetry 009-10: Questions 1. Using the definition of a group, prove the Rearrangement Theorem, that the set of h products RS obtained for a fixed element S, when R ranges over the h elements of the
On P Systems with Active Membranes
On P Systems with Active Membranes Andrei Păun Department of Computer Science, University of Western Ontario London, Ontario, Canada N6A 5B7 E-mail: apaun@csd.uwo.ca Abstract. The paper deals with the
FQXI 2014: Foundations of Probability and Information
FQXI 2014: Foundations of Probability and Information CONTINGENT or DERIVABLE To what degree are the Laws of Physics dictated yet arbitrary (i.e. must be discovered)? and to what degree are Laws of Physics
CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
SF2729 GROUPS AND RINGS LECTURE NOTES
SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking
On Weak Chromatic Polynomials of Mixed Graphs
On Weak Chromatic Polynomials of Mixed Graphs Daniel Blado California Institute of Technology Joseph Crawford Morehouse College July 27, 2012 Taïna Jean-Louis Amherst College Abstract Modeling of metabolic
Counting. Spock's dilemma (Walls and mirrors) call it C(n,k) Rosen, Chapter 5.1, 5.2, 5.3 Walls and Mirrors, Chapter 3 10/11/12
Counting Rosen, Chapter 5.1, 5.2, 5.3 Walls and Mirrors, Chapter 3 Spock's dilemma (Walls and mirrors) n n planets in the solar system n can only visit k
Isomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
Binary codes related to the moonshine vertex operator algebra
Binary codes related to the moonshine vertex operator algebra Akihiro Munemasa 1 1 Graduate School of Information Sciences Tohoku University (in collaboration with K. Betsumiya, R. A. L. Betty, M. Harada
Beginnings of Polya Theory
Beginnings of Polya Theory Naren Manjunath July 29, 2013 1 Introduction Polya theory is, unlike most of high- school combinatorics, not a bag of tricks that are situation- specific. It deals with questions
Math.3336: Discrete Mathematics. Combinatorics: Basics of Counting
Math.3336: Discrete Mathematics Combinatorics: Basics of Counting Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
Quantum Field theories, Quivers and Word combinatorics
Quantum Field theories, Quivers and Word combinatorics Sanjaye Ramgoolam Queen Mary, University of London 21 October 2015, QMUL-EECS Quivers as Calculators : Counting, correlators and Riemann surfaces,
The Geometry of Root Systems. Brian C. Hall
The Geometry of Root Systems A E Z S Brian C. Hall T G R S T G R S 1 1. I Root systems arise in the theory of Lie groups and Lie algebras, but can also be studied as mathematical objects in their own right.
arxiv:math/ v1 [math.co] 21 Jun 2005
![arxiv:math/ v1 [math.co] 21 Jun 2005](/thumbs/72/67889588.jpg "arxiv:math/ v1 [math.co] 21 Jun 2005")
Counting polyominoes with minimum perimeter arxiv:math/0506428v1 [math.co] 21 Jun 2005 Abstract Sascha Kurz University of Bayreuth, Department of Mathematics, D-95440 Bayreuth, Germany The number of essentially
MATHEMATICS Trigonometry. Mathematics 30-1 Mathematics (10 12) /21 Alberta Education, Alberta, Canada (2008)
MATHEMATICS 30-1 [C] Communication Trigonometry Develop trigonometric reasoning. 1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians. [CN, ME, R, V] 2. Develop
Axioms of Kleene Algebra
Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.
The Hypercube Graph and the Inhibitory Hypercube Network
The Hypercube Graph and the Inhibitory Hypercube Network Michael Cook mcook@forstmannleff.com William J. Wolfe Professor of Computer Science California State University Channel Islands william.wolfe@csuci.edu
Book announcements. Sukumar Das Adhikari ASPECTS OF COMBINATORICS AND COMBINATORIAL NUMBER THEORY CRC Press, Florida, 2002, 156pp.
Discrete Mathematics 263 (2003) 347 352 www.elsevier.com/locate/disc Book announcements Sukumar Das Adhikari ASPECTS OF COMBINATORICS AND COMBINATORIAL NUMBER THEORY CRC Press, Florida, 2002, 156pp. Notations
Individual Round CHMMC November 20, 2016
Individual Round CHMMC 20 November 20, 20 Problem. We say that d k d k d d 0 represents the number n in base 2 if each d i is either 0 or, and n d k ( 2) k + d k ( 2) k + + d ( 2) + d 0. For example, 0
Factoring. Expressions and Operations Factoring Polynomials. c) factor polynomials completely in one or two variables.
Factoring Strand: Topic: Primary SOL: Related SOL: Expressions and Operations Factoring Polynomials AII.1 The student will AII.8 c) factor polynomials completely in one or two variables. Materials Finding