BIJECTIVE PROOF PROBLEMS
August 18, 2009
Richard P. Stanley
The statements in each problem are to be proved combinatorially, in most
cases by exhibiting an explicit bijection between two sets. Try to give the
most elegant proof possible. Avoid induction,
Universal Cycles
2011
Yu She
Wirral Grammar School for Girls
Department of Mathematical Sciences
University of Liverpool
Supervisor: Professor P. J. Giblin
Contents
1 Introduction
2
2 De
2.1
2.2
2.3
2.4
2.5
2.6
2
2
5
6
6
7
9
Bruijn sequences and Eulerian
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
dominoes that each may cover two adjacent squares (ho
Lecture Notes
Combinatorics
Lecture by Torsten Ueckerdt
Problem Classes by Jonathan Rollin
Lecture Notes by Stefan Walzer Errors go here. Thanks!
Last updated: July 22, 2015
1
Contents
0 What is Combinatorics?
4
1 Permutations and Combinations
1.1 Basic C
THE q-SERIES IN COMBINATORICS;
PERMUTATION STATISTICS
(Preliminary version)
May 5, 2011
Dominique Foata and Guo-Niu Han
Guo-Niu Han
I.R.M.A. UMR 7501
Universite Louis Pasteur et CNRS
7, rue Rene-Descartes
F-67084 Strasbourg, France
guoniu@math.u-strasbg.f
Some simple graph spectra
The (ordinary) spectrum of a graph is the spectrum of its (0,1) adjacency
matrix. (There are other concepts of spectrum, like the Laplace spectrum or
the Seidel spectrum, that are the spectrum of other matrices associated with
th
Topics in generating functions
Qiaochu Yuan
Massachusetts Institute of Technology
Department of Mathematics
Written for the Worldwide Online Olympiad Training program
http:/www.artofproblemsolving.com
April 7th, 2009
1
Introduction
Suppose we want to stud
Counting spanning trees
Lector: Alexander Mednykh
Sobolev Institute of Mathematics
Novosibirsk State University
Winter School in Harmonic Functions on Graphs
and Combinatorial Designs
20 - 24 January, 2014
Mednykh A. D. (Sobolev Institute of Math)
Spannin
Math 296. Homework 1 (Due January 14, 2011)
The Cantor Problem Set: . . . the most astonishing product of mathematical thought, the most beautiful realization
of human activity in the domain of the purely intelligible. No one shall expel us from the parad
CSE 713: Random Graphs and Applications
SUNY at Buffalo, Fall 2003
Lecturer: Hung Q. Ngo
Scribe: Hung Q. Ngo
Lecture 10: Introduction to Algebraic Graph Theory
Standard texts on linear algebra and algebra are [2,14]. Two standard texts on algebraic graph
Advanced Graph Algorithms
Jan-Apr 2014
Lecture 24 April 23, 2014
Lecturer: Sanjukta Roy
1
Scribe: Sanjukta Roy
Overview
In the last presentation the lemma of Gessel-Viennot was presented. Today we will see the proof
of Tuttes Matrix Tree Theorem using the
May 25, 2002 12:26 WSPC/Guidelines
01167
International Journal of Modern Physics B, Vol. 16, Nos. 14 & 15 (2002) 19511961
c World Scientific Publishing Company
DIMERS AND SPANNING TREES: SOME RECENT RESULTS
F. Y. WU
Department of Physics, Northeastern Uni
Fractals, Graphs, and Fields
Franklin Mendivil
1. INTRODUCTION. One of the most amazing facets of mathematics is the experience of starting with a problem in one area of mathematics and then following the
trail through several other areas to the solution
Lecture 10
Proof of the Matrix-Tree Theorem
The proof here is derived from a terse account in the lecture notes from a course on
Algebraic Combinatorics taught by Lionel Levine at MIT in Spring 2011.1 I studied
them with Samantha Barlow, a former Discrete
Enumerative Combinatorics 5:
q-analogues
Peter J. Cameron
Autumn 2013
In a sense, a q-analogue of a combinatorial formula is simply another
formula involving a variable q which has the property that, as q 1, the
second formula becomes the first. Of course
Lecture 7
The Matrix-Tree Theorem
This section of the notes introduces a very beautiful theorem that uses linear algebra
to count trees in graphs.
Reading:
The next few lectures are not covered in Jungnickels book, though a few definitions
in our Section
18.312: Algebraic Combinatorics
Lionel Levine
Lecture 19
Lecture date: April 21, 2011
1
Notes by: David Witmer
Matrix-Tree Theorem
1.1
Undirected Graphs
Let G = (V, E) be a connected, undirected graph with n vertices, and let (G) be the
number of spanning
PARTITION BIJECTIONS, A SURVEY
IGOR PAK
Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented
in a different, sometimes unrecognizable way. Various extens
The Matrix-Tree Theorem
Evans Doe Ocansey (evans@aims.ac.za)
African Institute for Mathematical Sciences (AIMS)
Supervised by: Prof. Stephan Wagner
Stellenbosch University, South Africa
19 May 2011
Submitted in partial fulfillment of a postgraduate diplom
TOPICS IN ALGEBRAIC COMBINATORICS
Richard P. Stanley
Version of 1 February 2013
4
CONTENTS
Preface
3
Notation
6
Chapter 1
Walks in graphs
9
Chapter 2
Cubes and the Radon transform
21
Chapter 3
Random walks
33
Chapter 4
The Sperner property
45
Chapter 5
Gr
Notes on partitions and their generating functions
1. Partitions of n.
In these notes we are concerned with partitions of a number n, as opposed to partitions of a set.
A partition of n is a combination (unordered, with repetitions allowed) of positive in
De Bruijn Cycles for Covering Codes
Fan Chung and Joshua N. Cooper
Department of Mathematics
University of California, San Diego, La Jolla, CA
August 4, 2003
Abstract
A de Bruijn covering code is a q-ary string S so that every qary string is at most R sym
arXiv:math/9903025v2 [math.CO] 3 Apr 2000
Trees and Matchings
Richard W. Kenyon
James G. Propp
David B. Wilson
Laboratoire de Topologie
University of Wisconsin
Microsoft Research
Universite Paris-Sud
Madison, Wisconsin
Redmond, Washington
kenyon@topo.math
10
Hadamard Matrices
Hadamard Matrix: An n n matrix H with all entries 1 and HH > = nI is called a
Hadamard matrix of order n. For brevity, we use + instead of 1 and
Examples:
[+]
"
+ +
+
2
#
+ + + +
6
6 + +
6
6 +
+
4
+
+
instead of
1.
3
7
7
7
7
5
Notes:
Partitions, permutations and posets
Pter Csikvri
In this note I only collect those things which are not discussed in R. Stanleys
Algebraic Combinatorics book.
1. Partitions
For the denition of (number) partition, Ferrers diagram and Young tableaux,
conjug
"Professor Andrey" Mishchenko Math 296 Lecture Notes
Nick Wasylyshyn
January 24-28, 2010
January 28 : Complex Topology and Analysis
Last class period, there wasnt enough time to discuss everything that we should know about
roots of unity. We will touch on
"Professor" Andrey Mishchenko Math 296 Lecture Notes
Nick Wasylyshyn
January 24-28, 2010
January 25 : Complex Polynomials
Preliminary notes
There will be a quiz tomorrow ! Review notes and definitions and it will be easy.
What is a Polynomial over C ?
Def
"Professor" Andrey Mishchenko Math 296 Lecture Notes
Nick Wasylyshyn
January 24-28, 2010
January 26 : The Geometry of the Complex Numbers
Before we discuss the geometry of C, we should consider a few things about its algebra.
Two Last Algebraic Questions
"Professor" Andrey Mishenko Math 296 Lecture Notes
Nick Wasylyshyn
January 24-28, 2010
January 24 : An Introduction to Complex Numbers
Preliminary notes
There will be a quiz sometime this week. As with Prof. Smiths quizzes, as long as we know
the definiti