Combinatorics [Lecture notes]

192 Pages·2012·2.74 MB·english

by  N. G. de Bruijn

J. W. Nienhuys
de Bruijn’s
COMBINATORICS
Classroomnotes
Ling-Ju Hung and Ton Kloks (eds.)
March 20, 2012
In honor of
N. G. de Bruijn and J. W. Nienhuys.

Preface
These are Nienhuys’ lecture notes on a course given by de Bruijn starting in
thelate1970sandallthroughthe1980s.Wethoughtitwastimetotranslate
thisclassictextintoEnglish.
This text emphasizes the study of equivalences in enumeration methods.
Nowadays, much research in graph theory and graph algorithms focuses on
tree-decompositions of graphs. In order to obtain efficient algorithms using
thesetree-structures,agoodunderstandingofequivalentsolutionsandequiv-
alenttree-structuresareofimmenseimportance.Morethanothertextsinba-
sic combinatorics, this text emphasizes enumeration techniques and thereby
trains a student to recognize equivalent combinatorial objects. The center of
thebookisPo´lya’senumerationtheory.Thistheoryiswithoutdoubtthebest
developed general tool for the enumeration of, e.g., geometrical and chemi-
cal objects that are equivalent under certain group operations. The chapters
on generating functions and permutations slowly build up to the chapter on
Po´lya’s theory. The chapter on graphs further emphasizes enumerations of
trees under certain equivalences. The chapter on bipartite graphs turns the
attention in the direction of algorithmics. It emphasizes covering and repre-
sentativeproblems.
Games are widely studied in mathematics, economics, and computer sci-
ence. The chapter on games is a playful introduction, by various nim games,
intoGrundyfunctiontheory.
Translatingthistextbroughtoneofusbackintotheclassroom,listeningto
deBruijnexplaininghiscombinatorics.Thesenotesaretrueclassroomnotes;
the teachings of de Bruijn were written down verbatim by Nienhuys and the
blackboard material was copied in numerous figures. This gives the text a
unique,almostmagical,speedandclaritywhichonecannotfindinothertext
books.
The original text lacked exercises. Besides ‘ordinary’ exercises, we have
tried to incorperate various classic papers here, of e.g., van der Waerden,
Berge, Tutte and, of course, de Bruijn. The student is guided through these
beautifulpapersinastep-by-stepmanner.
VIII Preface
We taught this course in the spring-semester 2010 to undergraduate stu-
dents at the Department of Computer Science and Information Engineering
ofNationalChungChengUniversity,Ming-Shiun,Chia-Yi,Taiwan.Theenthu-
siasm with which our students welcomed this course surpassed our wildest
dreams.Itprovedtousthat,withoutdoubt,thepreservationofthistextwill
bringpleasuretomanygenerationsofcombinatoricsstudentstocome.
We are deeply indebted to professor Dan Buehrer for proofreading and
correctingourEnglishtranslation.
This project was supported by the National Science Council of Taiwan
undergrantsNSC97-2811-E-194-001andNSC99-2218-E-007-016.Wethank
theNationalScienceCouncilofTaiwanfortheirsupport.
Ling-JuHung
DepartmentofComputerScienceand
InformationEngineering
NationalChungChengUniversity
Ming-Shiun,Chia-Yi,621,Taiwan
TonKloks
DepartmentofComputerScience
NationalTsingHuaUniversity
Hsinchu,Taiwan.
Contents
1 Introduction................................................ 1
2 GeneratingFunctions........................................ 3
2.1 Smallchange ........................................... 6
2.2 Asummationformula .................................... 7
2.3 Multisets ............................................... 11
2.4 Theweighingproblem ................................... 13
2.5 Partitionsofnumbers .................................... 14
2.6 Notes .................................................. 17
2.7 Problems ............................................... 18
3 Permutations ............................................... 23
3.1 Theshepherd’sprinciple.................................. 23
3.2 Permutations............................................ 24
3.3 Stirlingnumbers ........................................ 27
3.4 Partitionsofsets......................................... 29
3.5 Arrangementsofmultisets ................................ 31
3.6 Numberedpartitions ..................................... 33
3.7 Problems ............................................... 34
4 Po´lyatheory ................................................ 39
4.1 Blackandwhitecubes.................................... 39
4.2 Cycleindex ............................................. 41
4.3 Cauchy-Frobeniuslemma................................. 42
4.4 Coloringsandcolorpatterns .............................. 45
4.5 Po´lya’stheorem ......................................... 46
4.6 Invariantcolorings....................................... 53
4.7 Superpatterns .......................................... 57
4.8 Wreathproduct ......................................... 64
4.9 Problems ............................................... 66
X Contents
5 Graphs ..................................................... 75
5.1 Introduction ............................................ 75
5.2 Moreintroduction ....................................... 76
5.3 Trees .................................................. 80
5.4 Pru¨fercode ............................................. 84
5.5 Countingtrees .......................................... 86
5.6 Cayley’sfunctionalequation .............................. 88
5.7 UD-encoding............................................ 94
5.8 KE-encoding ............................................ 97
5.9 Countingalcohols .......................................100
5.10 Thematrixtreetheorem..................................102
5.11 Eulergraph.............................................109
5.12 Linegraph ..............................................112
5.13 DeBruijnsequence ......................................114
5.14 Spanningtrees ..........................................122
5.15 Problems ...............................................124
6 Bipartitegraphs.............................................131
6.1 Introduction ............................................131
6.2 Covering–andrepresentativesproblem ....................133
6.3 TheoremofKo¨nigandHall ...............................134
6.4 TheoremofKo¨nigandEgerva´ry ...........................136
6.5 Latinsquares ...........................................138
6.6 TheoremofFord-Fulkerson ...............................139
6.7 Problems ...............................................144
7 Graphsandgames ..........................................159
7.1 Introduction ............................................159
7.2 Pay-offfunction .........................................161
7.3 Nim ...................................................163
7.4 Grundyfunction.........................................165
7.5 Wijthoffgame...........................................166
7.6 Othergames ............................................169
7.7 Problems ...............................................170
References .....................................................173
Index ..........................................................175
1
Introduction
N.G.deBruijngaveacourseinCombinatoricsduringthe1980satEindhoven
University of Technology. During the course J. W. Nienhuys took notes. This
isatranslationofthesenotes.
What is combinatorics? You could say that combinatorics is that part of
mathematics that concerns itself with finite systems. But this is not entirely
true:thetheoryoffinitegroups,ringsandfieldsdoesnotbelongtocombina-
torics.
Itiswithcombinatoricsasitiswithasymptotics.Youwouldliketodefine
asymptotics as the theory of limits, but large parts of the differential– and
integration-calculusdonotbelongtoasymptotics.
Foralongtimecombinatoricsconsistedonlyofpuzzlesandgames.Before
the1960stherewasprobablynocourseincombinatorics.
Combinatorics is mainly about counting. The nice thing about counting
is that you learn something about the thing that you are counting. While
counting,younoticethatyourknowledgeaboutthesubjectisnotsufficiently
precise,andsocountingbecomesaneducationalactivity.Whentwosetshave
the same number of elements, you try to understand why that is the case by
establishingsomenaturalbijectionbetweenthetwo.
N.G.deBruijnrecallsthatsometimearound1975,hecountedthenumber
ofacertaintypeoflogicswiththreevariables.Therewerealotofthem;some
number with 14 digits. A little while later he saw an article that was about
something completely different, but it concluded that the number of those
objectswasexactlythatsamenumberwith14digits.Hegotcuriousandread
thearticleverycarefully.Indeed,ifyouthoughtdeeplyaboutityoucouldsee
thatyoucouldinterpretthoseobjectsalsoaslogics.
Where is combinatorics applied? There is a lot of counting going on in
statistics;andincomputersciencecombinatoricsplaysaroleinconsiderations
aboutcomplexity.
Let’sjuststart,anddosomecombinatorics.

2
Generating Functions
Oneveryoldideaincombinatoricsistheideaofthegeneratingfunction.This
ideacanbetracedbacktoLaplace.
Givenaninfinitesequence
a ,a ,a ,...
0 1 2
of numbers, then the generating function of that sequence is the following
series:
A(x)=a +a x+a x2+...
0 1 2
Thetechniqueofthegeneratingfunctionisthatyoutrytofollowthenext
flowchart:
knowledge about translation knowledge
thesequence −−f−or−w−a−r−d→ aboutA(x)
operations

more knowl- 
translation more kynowledge
edge about
thesequence. ←b−ac−k−w−a−r−d−s aboutA(x)
ThisflowchartisabitlikethatoftheLaplaceintegral
F(x)= extf(t)dt.
Z0∞
Theanalogousflowchartis: