J. W. Nienhuys de Bruijn’s COMBINATORICS Classroom notes 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 the late 1970s and all through the 1980s. We thought it was time to translate this classic text into English. 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 these tree-structures, a good understanding of equivalent solutions and equiv- alent tree-structures are of immense importance. More than other texts in ba- sic combinatorics, this text emphasizes enumeration techniques and thereby trains a student to recognize equivalent combinatorial objects. The center of the book is Polya’s´ enumeration theory. This theory is without doubt the best 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 Polya’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- sentative problems. Games are widely studied in mathematics, economics, and computer sci- ence. The chapter on games is a playful introduction, by various nim games, into Grundy function theory. Translating this text brought one of us back into the classroom, listening to de Bruijn explaining his combinatorics. These notes are true classroom notes; 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, almost magical, speed and clarity which one cannot find in other text 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 beautiful papers in a step-by-step manner. VIII Preface We taught this course in the spring-semester 2010 to undergraduate stu- dents at the Department of Computer Science and Information Engineering of National Chung Cheng University, Ming-Shiun, Chia-Yi, Taiwan. The enthu- siasm with which our students welcomed this course surpassed our wildest dreams. It proved to us that, without doubt, the preservation of this text will bring pleasure to many generations of combinatorics students to come. We are deeply indebted to professor Dan Buehrer for proofreading and correcting our English translation. This project was supported by the National Science Council of Taiwan under grants NSC 97-2811-E-194-001 and NSC 99-2218-E-007-016. We thank the National Science Council of Taiwan for their support. Ling-Ju Hung Department of Computer Science and Information Engineering National Chung Cheng University Ming-Shiun, Chia-Yi, 621, Taiwan Ton Kloks Department of Computer Science National Tsing Hua University Hsinchu, Taiwan. Contents 1 Introduction ................................................ 1 2 Generating Functions ........................................ 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 Polya´ theory ................................................ 39 4.1 Blackandwhitecubes............................... ..... 39 4.2 Cycleindex........................................ ..... 41 4.3 Cauchy-Frobeniuslemma............................. 42 4.4 Coloringsandcolorpatterns ........................ ...... 45 4.5 Polya’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 Prufercode...........................................¨ . 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 5.11Eulergraph..................................... ........109 5.12Linegraph ....................................... .......112 5.13DeBruijnsequence ................................ 114 5.14Spanningtrees ................................... .......122 5.15Problems....................................... ........124 6 Bipartite graphs .............................................131 6.1 Introduction ..................................... .......131 6.2 Covering – and representatives problem ................ 133 6.3 Theorem of KonigandHall¨ ...............................134 6.4 Theorem of Konig¨ and Egervary´ ...........................136 6.5 Latinsquares ..................................... 138 6.6 TheoremofFord-Fulkerson ......................... 139 6.7 Problems........................................ .......144 7 Graphs and games ..........................................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. de Bruijn gave a course in Combinatorics during the 1980s at Eindhoven University of Technology. During the course J. W. Nienhuys took notes. This is a translation of these notes. 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: the theory of finite groups, rings and fields does not belong to combina- torics. It is with combinatorics as it is with asymptotics. You would like to define asymptotics as the theory of limits, but large parts of the differential– and integration-calculus do not belong to asymptotics. For a long time combinatorics consisted only of puzzles and games. Before the 1960s there was probably no course in combinatorics. Combinatorics is mainly about counting. The nice thing about counting is that you learn something about the thing that you are counting. While counting, you notice that your knowledge about the subject is not sufficiently precise, and so counting becomes an educational activity. When two sets have the same number of elements, you try to understand why that is the case by establishing some natural bijection between the two. N. G. de Bruijn recalls that sometime around 1975, he counted the number of a certain type of logics with three variables. There were a lot of them; 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 objects was exactly that same number with 14 digits. He got curious and read the article very carefully. Indeed, if you thought deeply about it you could see that you could interpret those objects also as logics. Where is combinatorics applied? There is a lot of counting going on in statistics; and in computer science combinatorics plays a role in considerations about complexity. Let’s just start, and do some combinatorics. 2 Generating Functions One very old idea in combinatorics is the idea of the generating function. This idea can be traced back to Laplace. Given an infinite sequence a0, a1, a2,... of numbers, then the generating function of that sequence is the following series: 2 A(x) = a0 + a1x + a2x + ... The technique of the generating function is that you try to follow the next flowchart: knowledge about translation knowledge the sequence −−−−−−−→forward about A(x) operations more knowl- translation more knowledge edge about about