EXERCISES IN
John McCleary (Mathematical)STYLE 10.1090/nml/050
Exercises in (Mathematical) Style
Stories of Binomial Coefficients Originally published by The Mathematical Association of America, 2017. ISBN: 978-1-4704-4783-0 LCCN: 2017939372
Copyright c 2017, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2018 The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 23 22 21 20 19 18 Exercises in (Mathematical) Style
Stories of Binomial Coefficients
John McCleary Vassar College
Providence, Rhode Island Council on Publications and Communications Jennifer J. Quinn, Chair Anneli Lax New Mathematical Library Editorial Board Karen Saxe, Editor Timothy G. Feeman John H. McCleary Katharine Ott Katherine Socha James S. Tanton Jennifer M. Wilson ANNELI LAX NEW MATHEMATICAL LIBRARY
1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F. Beckenbach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinoff 5. The Contest Problem Book I Annual High School Mathematics Examinations 1950–1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by P. J. Davis 7. Uses of Infinity by Leo Zippin 8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. Olds 10. Replaced by NML-34 11. Hungarian Problem Books I and II, Based on the Eötvös Competitions 12. 1894–1905 and 1906–1928, translated by E. Rapaport 13.o Episodes from the Early History of Mathematics by A. Aaboe 14. Groups and Their Graphs by E. Grossman and W. Magnus 15. The Mathematics of Choice by Ivan Niven 16. From Pythagoras to Einstein by K. O. Friedrichs 17. The Contest Problem Book II Annual High School Mathematics Examinations 1961–1965. Compiled and with solutions by Charles T. Salkind 18. First Concepts of Topology by W. G. Chinn and N. E. Steenrod 19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations II by I. M. Yaglom, translated by A. Shields 22. Elementary Cryptanalysis by Abraham Sinkov, revised and updated by Todd Feil 23. Ingenuity in Mathematics by Ross Honsberger 24. Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book III Annual High School Mathematics Examinations 1966–1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26. Mathematical Methods in Science by George Pólya 27. International Mathematical Olympiads—1959–1977. Compiled and with solutions by S. L. Greitzer 28. The Mathematics of Games and Gambling, Second Edition by Edward W. Packel 29. The Contest Problem Book IV Annual High School Mathematics Examinations 1973–1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione, and N. Shell 30. The Role of Mathematics in Science by M. M. Schiffer and L. Bowden 31. International Mathematical Olympiads 1978–1985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin 32. Riddles of the Sphinx by Martin Gardner 33. U.S.A. Mathematical Olympiads 1972–1986. Compiled and with solutions by Murray S. Klamkin 34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson 35. Exploring Mathematics with Your Computer by Arthur Engel 36. Game Theory and Strategy by Philip D. Straffin, Jr. 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger 38. The Contest Problem Book V American High School Mathematics Examinations and American Invitational Mathematics Examinations 1983–1988. Compiled and augmented by George Berzsenyi and Stephen B. Maurer 39. Over and Over Again by Gengzhe Chang and Thomas W. Sederberg 40. The Contest Problem Book VI American High School Mathematics Examinations 1989–1994. Compiled and augmented by Leo J. Schneider 41. The Geometry of Numbers by C. D. Olds, Anneli Lax, and Giuliana P. Davidoff 42. Hungarian Problem Book III, Based on the Eötvös Competitions 1929–1943, translated by Andy Liu 43. Mathematical Miniatures by Svetoslav Savchev and Titu Andreescu 44. Geometric Transformations IV by I. M. Yaglom, translated by A. Shenitzer 45. When Life is Linear: from computer graphics to bracketology by Tim Chartier 46. The Riemann Hypothesis: A Million Dollar Problem by Roland van der Veen and Jan van de Craats 47. Portal through Mathematics: Journey to Advanced Thinking by Oleg A. Ivanov. Translated by Robert G. Burns. 48. Exercises in (Mathematical) Style: Stories of Binomial Coefficients by John McCleary Other titles in preparation. Preface
. . . it is extraordinary how fertile in properties the triangle is. Every- one can try his hand. Treatise on the Arithmetical Triangle, Blaise Pascal
At the age of twenty-one, he wrote a treatise upon the binomial theorem, which has had a European vogue. On the strength of it he won the mathematical chair at one of our smaller universities, and had, to all appearances, a most brilliant career before him. (Sherlock Holmes about James Moriarty) The Final Problem, Sir Arthur Conan Doyle
Who has anything new to say about the binomial theorem at this late date? At any rate, I am certainly not the man to know. (Moriarty to Watson) The Seven-Per-Cent Solution, Nicholas Meyer
. . . every style of reasoning introduces a great many novelties, in- cluding new types of (i) objects, (ii) evidence, (iii) sentences, new ways of being a candidate for truth or falsehood, (iv) laws, or at any rate modalities, (v) possibilities. One will also notice, on occasion, new types of classification, and new types of explanations. Ian Hacking
According to the Oxford English Dictionary, a pastiche is a work that in- corporates several different styles, drawn from a variety of sources, in the style of someone else. In this book I am consciously imitating the work Exercices de Style (Éditions Gallimard, Paris, 1947) of Raymond Queneau (1903–1976). In a page, Queneau introduces a banal story of a man on a bus that he then tells in 99 different ways. The manners of speech include philosophical, metaphorical, onomatopoeic, telegraphic; also the forms of an advertisement, a short play, a libretto, a tanka; in passive voice, imperfect tense; and so on. He celebrates the potential of language, and his experiments in style, even the most demanding ones to read, seek to delight the reader with new forms.
vii viii Preface
Queneau was no stranger to mathematics, having published a note in the Comptes Rendus and a paper in the Journal of Combinatorial Theory (both on s-additive sequences). He also published Cent mille milliards de poèmes (100,000,000,000,000 Poems1), consisting of a set of ten sonnets with a uni- fied rhyme scheme from which a poem is constructed by making a choice of each line from among the corresponding lines in the ten sonnets. This scheme produces the 1014 distinct poems. The poet estimated that it would require more than 190 million years to recite them all. Queneau was also the cofounder in 1960 of the experimental writing group OuLiPo (Ouvroir de Littérature Potentielle). From their earliest meetings the OuLiPo discussed the use of mathematical structures in literary endeavors. When strict rules (constraints) are chosen and followed, a writer needs to find new modes of expression. By focusing on new operations applied to literary works or generating new works, writing can become freer and more thorough, a labo- ratory of invention. The canonical example is La Disparation by George Perec (1936–1982), Denoël, Paris, 1969, a novel in which the vowel e has vanished. Such a text is called a lipogram. Mathematicians and authors well known to the mathematical community have been members of the OuLiPo since its origin, including cofounder François Le Lionnais, Claude Berge, and Michèle Audin. So what have I tried to do? I have chosen a (not at all) banal object in mathematics—the binomial coefficients—and written 99 short notes in which these numbers (or their cousins) play some role. Certain particular properties of these numbers appear in different exercises, seen from different viewpoints. The behavior of the binomial coefficients provides a framework that suggests properties of related families of numbers. The binomial coefficients even appear as a sort of character in a mathematical story, or they structure the choices in a piece. The temptation to write a mathematical pastiche of Exercices de Style has been on my mind for a long time, and it has also been taken up by others: Ludmilla Duchêne and Agnès Leblanc have written the delightful Rationnel mon Q (Hermann, Paris, 2010) in which 65 exercises explore the irrationality of the square root of 2. What is style? According to Wikipedia, style is a manner of doing or pre- senting something. In mathematics, there may be an algebraic way to under- stand certain results, and there also may be a combinatorial way to understand the same fact. Thus viewpoint contributes to style. Other aspects of style might include the method of a proof, be it induction, recursion, or algorithm; or by contradiction, by the introduction of a fancy zero, or a conscious imitation of the proof of a different fact. On the other hand, the presentation of a mathemat- ical proof might take a particular form of discourse.
1 An English translation of Cent mille milliards de poèmes can be found in Oulipo Compendium, compiled by Harry Mathews and Alastair Brotchie. London: Atlas Press; Los Angeles: Make Now Press, 2005. Preface ix
There is precious little discussion2 of style in mathematics, even if math- ematicians seem to have strong opinions. At the least, mathematicians seek to imitate their most admired authors’ styles of discourse. Like Moriarty I cannot hope to say anything new about the binomial coef- ficients. I am sure however that something new can be found on some pages by most readers. The exercises are intended to explore some of the possible ways to communicate mathematical ideas, and to reveal how binomial coefficients give mathematicians plenty to talk about. In some of the titles and in the Style notes at the end of the book, I try to identify the features of mathematical think- ing, proving, and writing on which each piece is based. I am not immune to the enthusiasms of the OuLiPo. As a general constraint each piece is no longer than three pages. The communication is the thing, and I have tried to tell a little something in each exercise. Mathematics is one of humanity’s oldest and deepest arts. The challenge of communicating mathematical ideas and truths deserves some critical attention, if only for the love of it. Queneau intended his Exercices de Style to, in his words, “act as a kind of rust-remover to literature.” I share the same intentions with him for my exercises. And, quoting Queneau,
“If I have been able to contribute a little to this, then I am very proud, especially if I have done it without boring the reader too much.” Raymond Queneau3
How to use this book The book is too thin to be useful as a doorstop, and it will anchor few pages in the role of paperweight. I recommend reading bits at random. You will find that neighboring pages of any particular exercise may be related in topic and so, if that topic interests you, by all means explore. Of course, read with a pencil (or pen) at hand to scribble particular cases, missing steps, critical remarks, conjec- tures (the next Fermat?), comments for the author, or shopping lists, as the spirit moves you. If a fog of obfuscation surrounds any page, acquaint yourself with the Style notes found at the end of the exercises, containing helpful citations, some of the author’s zany intentions, and valuable clues to hidden messages and
2See, for example, Variations du style mathématique by Claude Chevalley, Revue de Méta- physique et de morale 42(1935) 375–384. A more extended discussion may be found in the ar- ticle of Paolo Mancuso on mathematical style for the Stanford Encyclopedia of Philosophy, at the website: ♣❧❛ ♦✳ ❛♥❢♦ ❞✳❡❞✉✴❡♥ ✐❡ ✴♠❛ ❤❡♠❛ ✐❝❛❧✲ ②❧❡✴. Also, for a wide ranging dis- cussion of style in mathematics see the notes from the Oberwolfach workshop, “Disciplines and styles in pure mathematics, 1800–2000”, 28.II–6.III.2010, organized by David Rowe, Klaus Volk- ert, Phillippe Nabonnand, and Volker Remmert, found at ✇✇✇✳♠❢♦✳❞❡✴♦❝❝❛ ✐♦♥✴✶✵✵✾. 3Quoted in Many Subtle Channels by Daniel Levin Becker, Harvard University Press, 2012. x Preface
inside jokes4. Prerequisites for the text are few. Most of the exercises can be read after a good high school mathematics education, some require an acquaintance with calculus, and a few are written in the language of undergraduate courses in analysis, algebra, even topology. I have marked the entries in the table of contents with a star if calculus is used, two stars if complex numbers, linear algebra or number theory is expected, and three stars if exposure to higher level courses might be helpful. Dear Reader, the styles of thinking and speaking in more advanced fields of mathematics are fascinating as well, and exercises at this level will give you a glimpse of what is ahead. You can check the Style notes for introductory texts in the relevant fields. Enjoy the offerings here in the spirit of play in which they were written.
Acknowledgements Thanks to Hamish Short who introduced me to Queneau’s Exercices de Style and whose enthusiasm for it has infected me. Anastasia Stevens, Robert Ro- nan, and Lilian Zhao spent the summer of 2014 helping me find more ways in which the binomial coefficients appear in mathematics, providing subject matter for the exercises. Nancy Fox, Curtis Dozier, and Natalie Friedman shared their rhetorical expertise with me. Thanks also to folks who read bits, sometimes big bits, of the manuscript and offered their help in making it better: Ming- Wen An, Cristina Ballantine, Matthias Beck, Jan Cameron, Natalie Frank, Jerry Furey, Anthony Graves-McCleary, David Guichard, Liam Hunt, Beth Malm- skog, Clemency Montelle, and Kim Plofker. The difficult work of the copy- editor was expertly done by Underwood Dudley whose prose I admire and whose advice improved this book. All remaining errors are mine. The edito- rial board of the MAA’s Anneli Lax New Mathematical Library series, Timothy G. Feeman, Katharine Ott, Katherine Socha, James S. Tanton, and Jennifer M. Wilson, led by the ever-encouraging Karen Saxe, provided considerable wisdom and close readings of various versions of the manuscript. My thanks to all of you. The book is better for your time and effort. The efforts of Bev Ruedi to meet my typesetting demands were most appreciated. Thanks also to Steve Kennedy, the MAA’s Acquisition Editor, who immediately got what I was trying to do.
4This sentence is a pangram—a sentence containing all of the letters of the alphabet. Contents
Preface ...... vii Notations ...... 1 Combinatorial thinking ...... 3 An algebraic relation ...... 5 Combinatorial consequences ...... 7 Downtown Carré ...... 9 Proof without words ...... 11 A geometric representation ...... 12 6 3 tetrads ...... 14 A footnote ...... 15 Fancy evaluations ...... 16 Careful choices ...... 18 An explicit formula ...... 20 Explicit counting ...... 22 History ...... 23 The diagonal club ...... 25 The Chu-Vandermonde identity ...... 27 Up and down ...... 29 Mind the gaps ...... 31 Recurrence ...... 33 Rabbits ...... 35 Repetition ...... 37
xi xii Contents
Left out ...... 39 Sets ...... 41 The Ramsey game ...... 43 The Principle of Inclusion and Exclusion ...... 46 Inversion ...... 49 Derangements ...... 51 Seating the visiting dignitaries ...... 54 Multinomials, passively ...... 57 Don’t choose, distribute ...... 59 Tanka** ...... 62 Ode to a little theorem** ...... 63 Divisibility by a prime** ...... 65 A far finer gambit** ...... 68 Congruence modulo a prime** ...... 71 Alchemy** ...... 74 Close reading** ...... 77 An algorithm to recognize primes** ...... 79 Shifting entries** ...... 81 DIY primes** ...... 83 Polynomial relations ...... 86 The generalized binomial theorem ...... 88 Four false starts ...... 90 Protasis-apodosis** ...... 93 Matrices** ...... 96 Bourbaki ...... 99 Generating repetitions ...... 102 Dialogue concerning generating functions ...... 104 Counting trees ...... 107 q-analogues ...... 109 Contents xiii
A breakthrough** ...... 112 Partitions of numbers ...... 115 Take it to the limit ...... 118 q-binomial theorem ...... 120 The quantum plane*** ...... 123 Scrapsheet ...... 126 Mathematical Idol ...... 129 Symmetries ...... 132 Sums of powers* ...... 135 Parts of proof ...... 138 Rings and ideals*** ...... 140 Mathmärchen*** ...... 142 The binomial distribution ...... 145 Wormhole points ...... 147 You can’t always get what you want ...... 150 Matt Hu, Graduate student ...... 152 Seeking successes ...... 155 TEXNO…AINIA* ...... 157 An area computation* ...... 158 Experimental mathematics ...... 161 Telescopes ...... 163 The Rechner ...... 166 Lipogram ...... 168 18th century machinations ...... 170 Recipe* ...... 173 Math talk* ...... 175 Averages and estimates ...... 178 Equality*** ...... 180 , by parts* ...... 183 xiv Contents
A macaronic sonnet* ...... 186 A hidden integral* ...... 187 Letter to a princess* ...... 189 Explicit Eulerian numbers ...... 192 Mutperation satticsits*** ...... 195 Review* ...... 197 Plus C *...... 200 At the carnival** ...... 203 Indicators** ...... 206 Tweets** ...... 209 Beautiful numbers** ...... 210 Reminiscences*** ...... 212 Winter journal** ...... 215 Cellular automata ...... 218 Tiling ...... 221 Lattice points* ...... 224 Afterlife* ...... 227 Matt Hu and the Euler caper*** ...... 230 Hypergeometric musings*** ...... 233 On the bus*** ...... 236 Style notes ...... 239 Index ...... 271 Style notes
Notations. Notation is an important aspect of good mathematics. The other no- n tations for the binomial coefficients, such as C.n; k/, Ck , or nCk, are clut- tered and they can lead to misinterpretations and obfuscation. The history of mathematical notation is a fascinating topic. Famous accounts of nota- tion can be found in Florian Cajori’s A History of Mathematical Notations, in two volumes (1928–1929), Open Court Press, and the recent book of Joseph Mazur, Enlightening Symbols: A Short History of Mathematical No- tation and Its Hidden Powers, Princeton University Press, 2014. Combinatorial thinking. This exercise presents an alternative way to think about binomial coefficients. We will see in other exercises how relations among binomial coefficients can be established by counting the choices of an un- ordered subset of a finite set. This exercise also offers a problem solving strategy: rather than tackling a large problem directly, consider smaller ver- sions of the problem first because they may be more manageable and the evident shape of a solution may emerge. An algebraic relation. If the binomial coefficients are associated to binomials (they are), then the algebraic properties of binomials can lead to relations among their coefficients. The relation of the theorem is named for Pascal who saw how fundamental is it. However, it was known long before Pas- cal’s time in several different cultures. (See the exercise History.) Algebraic n relations will lead to further relations among the . Summation notation k is explained in the exercise. The proof is an example of algebraic thinking. Combinatorial consequences. At the base of combinatorial arguments there is the rule of sum: The cardinality of the union of two disjoints sets is the sum of the cardinalities of each set; and the multiplicative principle. If the number of ways to do A is n and to do B is m, then the number of ways to do both A and B is nm. The rule of sum is applied in the proof of the theorem, and the multiplicative principle is applied in the proof of the proposition. Downtown Carré. The counting of paths in a lattice is a useful representation of certain combinatorial problems. In this exercise, the binomial coefficients arise naturally from the constraints on paths. The name Edwin is a nod to
239 240 Style notes
Edwin Abbott Abbott, the author of Flatland: A Romance in Many Dimen- sions, Dover Publications, Mineola, NY, 1992. Abbott’s main character is “A Square.” The argument here is a combination of previous two exercises. Proof without words. A proof without words is most successful when the viewer needs to spend a little time seeing how the proof emerges. To give you some words to understand this exercise, the first picture shows the arrangement of the 2-element subsets of 1; 2; : : : ; n 1 into a triangle in a system- atic manner from which the identityf is proved.C g The numbers introduced are the triangular numbers Tn. In the case of the pyramidal numbers Pn, each layer of the pyramid is a triangular number and the three-element subsets of 1; 2; : : : ; n 2 are pictured. The study of families of numbers associated fwith shapes,C figurativeg numbers, is ancient and signals the beginnings of number theory. A geometric representation. The representation of .a b/n for the exponent 2 C 2 is found in Euclid, Book II, Proposition 4. This appearance of 2 is 1 D sometimes cited as the first instance of a case of the binomial theorem (see J. L. Coolidge, The story of the binomial theorem, Amer. Math. Monthly 56(1949), 147–157). 6 tetrads. Label the C major scale C 1, D 2,..., B 7. Then a choice 3 D D D of three values from 2; 3; : : : ; 7 , together with C, forms a tetrad, four tones f 6 g sounded together. There are possible tetrads. All of them occur exactly 3 once in some spelling in this piece, which, in the first two lines, is a canon at distance one on the familiar Dies Irae. A footnote. The letters, Epistola prior and Epistola posterior (see Correspon- dence of Isaac Newton, edited by H.W. Turnbull, Cambridge University Press, 1977, volume 2), were written long after Newton had deduced the general binomial theorem for rational exponents (see the exercise Experi- mental mathematics). In response to the curiosity of Leibniz, he described the manner in which he discovered the general theorem, giving enough hints to the power of his methods, and the scope of their application. Leibniz had discovered many of the same results in the Epistolae independently. New- ton’s observation about the powers of 11 is truly a footnote to the rest of the concepts discussed in these famous letters. Other uses of the data provided in Pascal’s triangle are possible along the lines suggested in the footnote; for example, powers of 12. The reader is encouraged to find their own co- incidences. Fancy evaluations. Algebraic expressions represent functions that may be eval- uated for special values of the variables. The reader can derive an infinite n n number of delightful relations in this manner. For example, 2k k 0 k D D X Style notes 241
3n. Polynomials represent a deeper notion, however. They can be under- stood as formal sums that are determined by their coefficients. In this frame- work, two polynomials are equal when their coefficients are equal. This principle leads us to compare like coefficients for polynomials that are equal for different reasons. For a discussion of the algebra of polynomials see, for example, Serge Lang’s book Undergraduate Algebra, Springer-Verlag, 3rd ed. 2005. Careful choices. In this exercise we prove by combinatorial means the same re- lations found in the exercise Fancy evaluations. The combinatorial method sets up a correspondence between the collection of things to be counted with one property and another collection of things with another property. When the correspondence is perfect, that is, a one-one correspondence, as in a subset and its complement, then the numbers of things in each collec- tion are seen to be the same. Such proofs are called bijective proofs. Bi- jective proofs are sometimes thought to be more “satisfying” as arguments than algebraic manipulations. Compare the exercises Careful choices and n n k Fancy evaluations and decide what you prefer. The relation k p D n n p is sometimes called the subset-of-a-subset identity. p k An explicit formula. The absorption identity is a consequence of the evaluation of another identity, the subset-of-a-subset identity. In this exercise we ex- plore how general relations can be evaluated to give particular cases that may open up new relations. Explicit formulas are not immediately available when numbers are defined algebraically or combinatorially. Arguments like the one found in this exercise are often needed to discover explicit formulas. Explicit counting. The relationship between ordered choices and unordered sub- sets leads to the explicit formula for binomial coefficients. The multiplica- tive principle plays a key role. To test your understanding, work out the ordering of the numbers of significant poker hands (pair, two pair, three of a kind, straight, flush, full house, four of a kind, straight flush): the fewer the instances, the better the hand. History. My historical remarks are based on the account found in The History of Mathematics by David Burton, 3rd edition, McGraw-Hill Companies, Inc., New York, NY, 1997. The reference to the table of al-Samaw’al comes from William Casselman, Binomial coefficients in Al-Bahir¯ f¯ı Al-Jabr, No- tices of the Amer. Math. Soc., 60(2013), 1498. My thanks to Jeff Suzuki, Clemency Montelle, and Kim Plofker for references on Indian mathemat- ics. In particular, the account of the work of Pingala˙ is based on the pa- per Sanskrit Prosody, Pingala˙ Sutras¯ and binary arithmetic by Ramaiyengar Sridharan, in Contributions to the History of Indian Mathematics, edited by Gérard G. Emch, Ramaiyengar Sridharan, and M. D. Srinvas, Hindustan 242 Style notes
Book Agency, 2005, Gurgaon, India. The diagonal club. This exercise refers to the fictional place found in Downtown Carré. The connection between binary digits and subsets is another way to sum the binomial coefficients. For a set like a; b; c; d , a binary number, say 0101, corresponds to the subset b; d . Generally,f g if you allow binary expressions to have zeroes in front tof getg them to length M , then there is a correspondence between subsets of a set with M elements and binary strings of length M . Counting in this manner was known to Pingala˙ in his treatise on poetic meters from 200 BCE. The Chu-Vandermonde identity. What appears in Chu Shih-Chieh’s Precious Mirror of the Four Elements is the identity
n r.r 1/ .r p 1/ n 1 p/.n 2 p/ .n q p/ C C C C C pŠ qŠ r 1 XD n r.r 1/ .r p q 1/ C C C ; D .p q/Š r 1 XD C which can be massaged into a version of the theorem. Joseph Needham (1900–1995), in his classic Science and Civilisation in China, Cambridge University Press, Cambridge, 1954, states that the identity is obtained in a discussion of series and progressions. Vandermonde proved the identity in his Mémoire sur des irrationnelles de différents ordres avec une applica- tion au cercle (1772). His paper treats in great detail the properties of falling factorials, .x/k x.x 1/ .x k 1/. The identity appears on p. 492 of his Mémoire.D Vandermonde was a medicalC doctor, violinist, and began contributing to mathematics at age 35. He is best known for his work on determinants. His arguments are developed further in the exercise 18th cen- tury machinations. Three proofs of the identity are found in this exercise: the first algebraic, the second combinatorial, and the third based on lattice path counts. Up and down. Limericks are a form of light poetry, usually based on the the anapest rhythm (two unaccented syllables followed by an accented one), and usually having three stressed syllables in lines 1, 2, and 5, and two stressed syllables in lines 3 and 4. The rhyme scheme is always AABBA. Limericks of a mathematical sort abound. My favorite ones refer to the Möbius band. One of the best I discovered is by Leigh Mercer and appeared in Word Ways, 13(1980), 36. Mathematically it is:
12 144 20 3p4 C C C .5 11/ 92 0: 7 C D C As a limerick it is: Style notes 243
A dozen, a gross, and a score, Plus three times the square root of four, Divided by seven Plus five times eleven Is nine squared and not a bit more. A function f .x/ is unimodal if there is a value m for which f .x/ is increasing when x m, and decreasing for x m. Hence f .m/ is a maximum value of f .x/. The unimodal property of binomial coefficients is shared by many other important combinatorial sequences. For a given func- tion or sequence, proving that it is unimodal can be difficult. Unimodality for distributions plays a role in probability theory. Mind the gaps. The general formula for the number of possible distributions to k individuals (numbered 1 to k) of n identical objects where each in- n 1 dividual gets at least one object is given by . The argument is k 1 exactly how Amalie describes it in the special case. For the number of pos- sible distributions of n identical objects to k individuals where it is possible n k 1 for individuals to get nothing, Hermann’s argument gives C .I k 1 imagine the scene of this exercise in an early 20th century German home, in Göttingen perhaps. Also see the exercise Repetitions. Recurrence. To define a sequence by a recurrence relation is a kind of axiomatic reduction of the sequence to a simpler form. The binomial coefficients were presented by Pascal in this form, leading to the arithmetical triangle and many of the results he obtained. Recurrence relations play an important role in computer science, offering, on one hand, a recipe for programming a computation, and on the other hand, straightforward descriptions of se- quences that test the limits of computation. Rabbits. Both sequences, of binomial coefficients and of Fibonacci numbers, are defined by recurrence relations, where each term is a sum of previous terms. A relation between the two sequences is not unexpected. The title of the exercise refers to the original problem posed by Fibonacci (Leonardo of Pisa (c 1170–c. 1250)). There is also a melding of the two notions called n fnfn 1 fn k 1 fibonomials: let C . The reader might enjoy de- k F D fkfk 1 f2f1 riving the properties of the fibonomials as modeled by the behavior of bino- n 1 n mial coefficients. For example, by proving C fn k 2 k F D C k F C n n fk 1 , it follows that is an integer for all n and k. See, k 1 F k F D. K. Hathaway, S. L. Brown, Fibonacci powers and a fascinating triangle, College Math. J. 28(1997), 124–128. 244 Style notes
Repetition. The theorem is another example of an recurrence relation that leads to an explicit closed form, with the help of Pascal’s identity. The form of the recurrence relation is sufficient for the construction of a table. The induction proof is more subtle being an induction on a sum of variables rather than a single variable. The odd discourse is inspired by Queneau’s exercise En partie double. Left out. The proof of this identity is a combinatorial argument and employs the ordering of the set of elements being counted to make distinctions between subsets. This artifact, that you can label the set of N distinct objects by the ordered labels 1 through N , provides a little extra feature that swings the argument. Furthermore, thinking about what is missing often can be as powerful as thinking about what is there. Sets. The arguments in this exercise are sometimes called bijective: To count the objects in a set, find another set whose cardinality is easier to determine and then construct a bijection between the set you want to count and the set with known cardinality. The notation emphasizes which sets are involved. If we X remove the assumption that X is finite, it is possible to define A Y D f X X #A #Y . We can ask how the set and its cardinality behave. j D g Y For example, if X N Y , the subsequent set is the collection of all countably infinite subsetsD ofD the natural numbers, an uncountable set. The Ramsey game. The arguments here are based on the well developed and sub- tle study of Ramsey theory, first considered in the paper of Frank P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30(1930), 264– 286. By recasting the basic theorems of Ramsey theory as a game, players can gain a hands-on feel for the combinatorial consequences of these ideas. A clear and thorough exposition of Ramsey theory can be found in Ron L. Graham, Bruce L. Rothschild, Joel H. Spencer, Ramsey Theory, Wiley and Sons, New York, NY, 1980. The game is best played on graph paper with pens of different colors. The Principle of Inclusion and Exclusion. The principle of inclusion-exclusion is one of the fundamental tools in combinatorics. In the sieve formulation, for example, to count primes, these ideas were known in antiquity. It is not clear when a “first” formal statement of the principle comes into the literature. For the probability version of the principle, it appears in Henri Poincaré’s Calcul des Probabilités, Gauthier-Villars, Paris, 1896. My account is based on a chapter of the excellent book, Combinatorics: Topics, Techniques, Al- gorithms by P. J. Cameron, Cambridge University Press, Cambridge, UK, 1994. Inversion. Binomial inversion links together certain sequences of numbers in un- expected ways. Applications can be found by expressing a known sequence Style notes 245
in terms of an unknown sequence and then applying inversion to obtain the unknown sequence. I apply this method in a few of the exercises to come. John Riordan’s excellent book, Combinatorial Identities, Wiley and Sons, New York, NY, 1968, devotes two chapters to inversion. When viewed prop- erly, inversion is an example of the principle of inclusion and exclusion, and even an example of matrix inversion (see the exercise Matrices).
Derangements. The problem of counting derangements is also known as the problème des rencontres, and it was first posed in 1708 by Pierre Rémond de Montmort in his Essai d’analyse sur les jeux de hazard. He solved the problem in 1713 in the third edition of the book. The argument for derange- ments is a beautiful example of the method of inversion, where the known sequence nŠ is related by inversion to the unknown sequence Dn . In- version recoversf g the unknown sequence in terms of the known. Byf theg way, Dn is the integer closest to nŠ=e.
Seating the visiting dignitaries. This combinatorial puzzle is a lightly disguised version of the well known problème des ménages, posed in 1891 by Édouard Lucas in his book Théorie des Nombres, Paris: Gauthier-Villars (pp. 491– 495) and solved in the decade following by Charles-Ange Laisant, M. C. Moreau, and H. M. Taylor. The original problem seats alternating husbands and wives around a circular table with no husband sitting next to his wife. I have chosen a non-sexist formulation with the visiting council. A related problem had been considered by Arthur Cayley (1821–1895) in 1878 that was shown to be equivalent by Jacques Touchard, Sur un prob- lème de permutations, C. R. Acad. Sciences Paris 198(1934), 631–633. It is Touchard who gave the formula at the end of the exercise. The proof presented here follows the classic proof of Irving Kaplansky, Solution of the ‘Problème des ménages’, B. Amer. Math. Soc., 49(1943), 784–785. An- other non-sexist version is given by Kenneth Bogart and Peter G. Doyle, Non-sexist solution of the ménage problem, Amer. Math. Monthly 93(1986), 514–519.
Multinomials, passively. This exercise features an example of the method of gen- eralization. We have some experience with binomial coefficients, so let’s extend it to more variables. For example, because the binomial coefficients satisfy the basic identity of Pascal, we can mimic a proof of that relation in the new context to see if it gives a relation for multinomial coefficients. (It does!) The multinomial theorem was discovered around 1676 by Leibniz but never published by him. De Moivre published a proof of it indepen- dently 20 years later. (See N. Bourbaki, Elements of the History of Mathe- matics, Springer-Verlag, Berlin, 1998.) If the tone of the exercise seems a bit formal, it is because the passive voice is something to be avoided. 246 Style notes
n Don’t choose, distribute. The generalized binomial coefficients were first k m studied by Abraham de Moivre (1667–1754) in The Doctrine of Chances: or, A method of calculating the probabilities of events in play (third edition 1756, printed for A. Millar, London), the first textbook on probability. De Moivre analyzed outcomes of games of chance and needed such coefficients for counting. Leonhard Euler made a thorough study of these numbers in De evolutione potestatis polynomialis cuiuscunque .1 x x2 x3 x4 etc./n, Nova Acta Academiae Scientarum ImperialisC C PetropolitinaeC C 12(1801),C 47–57. The four results of the exercise are established by a com- binatorial approach. The algebraic results in this exercise are parallel to the combinatorial arguments in the exercise Four false starts. Tanka/Haiku. The tanka is a form of Japanese poetry consisting of five lines, with each line containing a restricted number of syllables according to the pattern 5-7-5-7-7. The more familiar haiku has three lines, numbering 5-7-5 in syllables. If you are given k consecutive integers, they may be written n, n 1,..., n k 1. The product, divided by kŠ is given by C C n.n 1/ .n k 1/ n k 1 C C C : kŠ D k Because binomial coefficients are always integers, kŠ divides n.n 1/ .n k 1/. This short proof and the nontrivial nature of the resultC deserve theC interjection