EXERCISES IN

John McCleary (Mathematical)STYLE 10.1090/nml/050

Exercises in (Mathematical) Style

Stories of 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 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 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 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 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,

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 (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 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 (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 , 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 , 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 ...... 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 ...... 145 Wormhole points ...... 147 You can’t always get what you want ...... 150 Matt Hu, Graduate student ...... 152 Seeking successes ...... 155 TEXNO…AI€NIA* ...... 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 * ...... 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 . 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 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 , 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 , 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 , 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 , .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 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 . 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 . 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 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 (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. 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 , they may be written n, n 1,..., n k 1. The , 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 Wow! Ode to a little theorem. For Pierre de Fermat (1601–1665) and his contempo- raries Fermat’s little theorem would be stated: for any integer n and prime number p, p divides np n. In the notation of Gauss (1777–1855), intro- duced in his Disquistiones Arithmeticae, one would write np n.mod p/. p 1  If p does not divide n, then n 1.mod p/. The proof presented here follows ideas of Fermat; the more familiar proofs emphasize the algebraic properties of arithmetic modulo p. Induction was developed into a subtle tool by the mathematicians of Fermat’s time. The consequences of Fermat’s insight, Gauss’s notation, and the little theorem are far-reaching in modern elementary number theory (it deserves an ode). My favorite books on ele- mentary number theory include: G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers (sixth edition), Oxford University Press, 2008, New York, NY; Trygve Nagell, Introduction to Number Theory, American Mathematical Society; 2nd Chelsea Reprint edition (2001); and Underwood Dudley, Elementary Number Theory, Dover Publications; Second edition (2008), Mineola, NY. Divisibility by a prime. I based this exercise on the account of Kummer’s the- orem by Paulo Ribenboim in The New Book of Prime Number Records, Springer Verlag, NY, third edition 1996, and the e-survey of Andrew Style notes 247

Granville, Arithmetic properties of binomial coefficients, at ✇✇✇✳❞♠✳ ✉♠♦♥❡❛❧✳❝❛✴ ❛♥❞❡✇✴❇✐♥♦♠✐❛❧✴. Kummer’s result appeared in the paper Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. reine angew. Math. 44(1852), 93–146, where Kummer analyzed the be- havior of cyclotomic sums in order to prove a generalization of the law of quadratic reciprocity. The use of p-adic representations of integers is natu- ral in this situation. Also the modest process of carrying in addition takes on considerable significance in the analysis of divisibility. Nothing we learn, even from elementary school, is too modest to be useful.

A far finer gambit. The title of this exercise is taken from G.H. Hardy’s A Math- ematician’s Apology, Cambridge University Press, Cambridge, UK, 1940, in which he wrote “The proof is by reductio ad absurdum, . . . one of a math- ematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathe- matician offers the game.” The method of reductio ad absurdum or proof by contradiction is an important tool for fashioning proofs. The best known ex- amples include Euclid’s proof of the infinitude of primes, and the arithmetic proof of the irrationality of p2. The study of squarefree binomial coeffi- cients was initiated by Erdos˝ in Pal Erdos˝ and Ron L. Graham, Old and New Problems and Results in Combinatorial Number Theory, L’Enseignement 2m Math. Monographie 28, Genève, 1980. There Erdos˝ conjectured that m is squarefree for m > 4. This was proved for m > n0 for a large value n0 by A. Sárkozy,˝ On divisors of binomial coefficients. I., J. Number Th. 20(1985), 70–80. A complete proof following Sárkozy’s˝ ideas was given by A. Granville, and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43(1996), 73– 107. The lemma and theorem of the exercise, and the arguments, are from their paper. The authors sharpen the understanding of squarefree binomial coefficients, proving that “on average, there are approximately ten-and-two- thirds squarefree entries in a row of Pascal’s triangle.”

Congruence modulo a prime. This application of Fermat’s little theorem is par- ticularly satisfying, revealing the power of modular arithmetic. I chose a two-column proof for reasons of nostalgia. The results here first appeared in the papers É. Lucas, Sur les congruences des nombres eulériens et les coefficients différentiels des functions trigonométriques suivant un module premier, Bull. de la Soc. Math. de 6(1878) 49–54; J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime mod- ulus, Quart. J. of Pure and Applied Math. 30(1899), 150–156; and N. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54(1947) 589– 592. 248 Style notes

Alchemy. This exercise is based on the first part of the paper On certain prop- erties of prime numbers Quarterly J. of Pure and Applied Math. 5(1862), 35–39) by the Reverend Joseph Wolstenholme (1829–1891). I have taken some liberties in the derivation (for example, a little calculus) of the for- n n j 1 1 mula for Hn. Wolstenholme shows how the sum . 1/ C j 1 j j D 1 X   has a first difference, as a function of n, of . This does not suggest how he n came up with the formula. One lesson to learn from this exercise is that sim- plification of expressions can hide potential riches. Throughout the deriva- tion of the divisibility result we have chosen to consider a more compli- cated expression. A motto might be: “Simplify only if it leads to something richer.” The reference to Oresme and the harmonic series may be found in Quaestiones super geometriam Euclidis, edited by H.L.L. Busard, Leiden 1961; corrected version: J. E. Murdoch, Nicole Oresmes “Quaestiones su- per geometriam Euclidis,” Scripta Mathematica 27(1964), 67–91. I aimed for an alchemical tone based on readings from The Alchemists by F.S. Tay- lor, Henry Schuman, NY,1949, especially the passage attributed to Nicholas Flammel (pp. 163–173). Close reading. The exercise focuses on the explicit forms of the coefficients 3 c1 and c2 whose divisibility properties lead to the divisor of p . Together with Alchemy, it completes my close reading of Wolstenholme’s 1862 paper. Other proofs of Wolstenholme’s theorem may be found using methods such as Lucas’s theorem; for example, see J.W.L. Glaisher, On the residues of the sums of products of the first p 1 numbers, and their powers, to modulus p2, or p3, Quart. J. Math. 31(1900), 321–353. The remarkable review paper of Andrew Granville, Arithmetic properties of binomial coefficients. I. Bi- nomial coefficients modulo prime powers, in Organic Mathematics (Burn- aby, BC, 1995), CMS Conf. Proc., vol. 20, American Mathematical Society, Providence, RI, 1997, 253–275, contains many recent results of Wolsten- holme type. See also R. Me¸storovic,´ Wolstenholme’s theorem: Its gener- alizations and extensions in the last hundred and fifty years (1862–2012), 2p 1 arXiv:1111.3057v2. These papers include a relation between and p 1 the Bernoulli numbers and the latest news on the search for Wolstenholme  primes. An algorithm to recognize primes. One of the most desirable results in math- ematics is a necessary and sufficient condition for a positive integer to be prime. The best known example is Wilson’s theorem n 2 is prime if and only if n divides .n 1/Š 1:  C Babbage’s paper, Demonstration of a theorem relating to prime numbers, Edinburgh Philos. J. 1(1819), 46–49, contains the theorem on which the Style notes 249

2p 1 algorithm is based. It also contains the result that p2 divides if p p 1 is a prime. The Chu-Vandermonde identity plays a useful role in his proofs. 2p Think about 2. The pseudocode brings up the question of finding p   n an efficient algorithm to compute given n and k. Such an algorithm k would be needed to analyze the efficiency  of this primality test. Shifting entries. The paper of Mann and Shanks is very readable and gives a slightly different argument. In their paper the authors discuss their moti- vation for proving the result. They were studying the product over primes 1 1 1 1 . The idea of Granville mentioned in the proof may p 2 p2 p3 D Ybe found in Arithmetic properties of binomial coefficients. I. Binomial co- efficients modulo prime powers, in Organic Mathematics (Burnaby, BC, 1995), CMS Conf. Proc., vol. 20, American Mathematical Society, Provi- dence, RI, 1997, 253–275. DIY primes. I first learned Erdos’s˝ proof of Bertrand’s postulate in Wacław Sier- pinski’s´ Elementary theory of numbers, translated from Polish by A. Hulan- icki. Warszawa, Panstwowe´ Wydawn. Naukowe, 1964. The proof is a tour- de-force of elementary ideas, all in the right place. I hope you will dive in and fill in all the details for yourself. My Do-It-Yourself kit is based on the lovely exposition of the proof in Proofs from the BOOK by Martin Aigner and Günter Ziegler, 4th edition, Springer-Verlag, Berlin, 2010. Pál Erdos˝ was one of the most influential and productive mathematicians of the twen- tieth century. He would speak of the BOOK in which God has collected the perfect proofs of mathematical theorems. In 1932, aged 19, Erdos˝ published the paper Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5(1930-32), 194–198, in which he proved Bertrand’s postulate as outlined in the exercise—certainly a proof to be found in the BOOK. Polynomial relations. The main tool of this exercise, the lemma on polynomi- als and their shared values, can be used in ingenious ways to prove many relations. This useful fact can be applied, for example, when studying the geometric properties of polynomial functions, that is, in the study of al- gebraic geometry. Among the generalizations of binomial coefficients, the x polynomials play an important role, interpolating between the integer k values for x. See the exercise 18th century machinations for a development of such ideas. The generalized binomial theorem. It is an attractive aspect of mathematics that questions like What if? can be followed on by exploring as if an assertion is true to find out some of the consequences of a statement before proving it. Famously, Hardy and Littlewood explored the consequences of the failure 250 Style notes

of the Riemann Hypothesis, a failure that they thought would not be a bad thing for number theory. The proof in the exercise is based on the proof of Auguste Cauchy (1789–1857) of Newton’s binomial theorem found in his 1821 Cours d’analyse algébrique. The parts of the proof in the exer- cise pass muster in the present day regard for rigor. His assertion that the functions f .r; x/ are continuous because they are sums of continuous func- tions, however, does not meet this standard, as was pointed out by Abel in his remarkable 1826 paper on the . See the related exercises Equality and Afterlife. Four false starts. The title of this exercise is taken from Forty-one false starts: essays on artists and writers by Janet Malcolm (Farrar, Straus and Giroux, New York, 2013). False starts are a way of life in mathematics. The reasons for abandoning a course of development are sometimes computational: The right expression for a coefficient may depend on what role the numbers are n going to play. These generalized binomial coefficients were initially k m studied in the first textbook on probability by Abraham de Moivre in The Doctrine of Chances. De Moivre analyzed outcomes of games of chance for which he needed such coefficients for counting. Euler made a thor- ough study of these numbers in De evolutione potestatis polynomialis cuius- cunque .1 x x2 x3 x4 etc./n, Nova Acta Academiae Scientarum ImperialisC PetropolitinaeC C C12(1801),C 47–57. An interesting recent preprint by N.-E. Fahssi, Polynomial triangles revisited (arXiv:1202.0228v7) gives several interpretations of the Pascal-de Moivre triangle, including connec- tions to physics. Protasis-apodosis. The grammatical terms protasis and apodosis refer to the antecedent and consequent, respectively, of an if-then sentence. From the Greek, protasis is to stretch before, and apodosis, to give back. Grammati- cal implications can vary with the language. I learned of these terms from a discussion of the omen literature of ancient Babylonia in Clemency Mon- telle, Chasing Shadows: Mathematics, Astronomy, and the Early History of Eclipse Reckoning, John Hopkins U. Press, Baltimore, MD, 2011. Another example of this form of discourse is found in the children’s book, If You Give a Mouse a Cookie by Laura Joffe Numeroff and Felicia Bond, Harper Collins, New York, 2010. This exercise reveals some of the power of lin- ear algebra to organize spaces of polynomials. The Stirling numbers of the first and second kind are introduced and their basic properties derived. For a fascinating history of the Stirling numbers and more of their properties, see the article of Khristo N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag. 85(2012) 252–266. Matrices. Viewing the Pascal triangle as a matrix is a natural temptation and it leads to new formulations of the properties of the binomial coefficients. For Style notes 251

example, the existence of an inverse provides a new look at the method of binomial inversion. When studying matrices, certain operators on the ma- trices can be defined, and in this case we find the shift operator and the of Kemeny. The relation P P 0 is a different statement of the Pascal identity, which invites us to considerD other compartments of mathe- matics (analysis in this case). Matrix and operator proofs of combinatorial sums may be traced back to work of Leonard Carlitz, On arrays of num- bers, Amer. J. Math. 54(1932), 739–752. More general systems that result from infinite matrices like P have been investigated by others under the name of Riordan arrays (see R. Sprugnoli, Riordan arrays and combina- torial sums, Discrete Math. 132(1994), 267–290), and Galton arrays (see E. Neuwirth, Recursively defined combinatorial functions: extending Gal- ton’s board, Discrete Math. 239(2001), 33–51. Bourbaki. Nicholas Bourbaki, the author of the celebrated Éléments de Mathe- matique, is unjustly accused of a dry and pedantic style of exposition. This tone may be the result of the joint nature of the writing. Bourbaki was a group of young French mathematicians, who first met together in 1934 with the goal of producing a new and practical curriculum for analysis in the French universities. The project and its history are told in the book of Mau- rice Mashaal, Bourbaki: A Secret Society of Mathematicians. Translated by Anna Pierrehumbert. Providence, RI: Amer. Math. Soc., [2002] 2006. The analysis of mappings and the language of surjections, injections, and bijec- tions was developed in their work. Bourbaki never published an account of les artes combinatoires. The symbol “dangerous bend,” adopted from road- side signs, appears in the works of Bourbaki to indicate difficult aspects of an argument. The numbers S.n; k/ #SUR.Œn; Œk/=kŠ are, in fact, the same as the Stirling numbers of the secondD type that appear in the exercise Protasis-apodosis. The reader may enjoy exploring the equivalence of these counts. Generating repetitions. Generating functions and the algebra of formal provide parts of the foundation of the study of combinatorics. The economy of expression, ease of manipulation, and sometimes startling re- lations revealed by generating functions make them a very attractive object of study. See the classic book by Herb Wilf, generatingfunctionology, 3rd edition, AK Peters, 2006, Wellesley, MA. Questions of convergence of for- mal power series are not considered, only their algebraic structure. Equal- ity in this context is equality of formal power series—like powers have equal coefficients—an algebraic criterion that does not require evaluation at a value of x to produce an equation. Dialogue concerning generating functions. The Catalan numbers 1 2n 2n 2n Cn D n 1 n D n n 1 C       252 Style notes

were introduced by Euler to count the number of ways to divide a poly- gon into triangles. The sequence is named after Eugène Charles Catalan (1814–1894), who found that the values count the number of ways one can introduce parentheses to a product of (non-associative) variables and ob- tain a meaningful product. The sequence also appears in the work of the Chinese mathematician Ming’antu (c.1692–c.1763) on infinite series ex- pressions for sin.2x/ in terms of sin.x/. The ubiquity of Catalan numbers and their properties can be explored in the recent book Catalan Numbers by Richard P. Stanley, Cambridge University Press, New York, 2015. This ex- ercise presents the standard derivation of the values of the Catalan numbers. It illustrates the power of generating functions to obtain explicit formulas, even for complicated recurrences. I particularly like how the quadratic for- mula makes an appearance. The binomial theorem for rational exponents gives the needed infinite series. The title and discourse are based on Plato’s Meno and on Galileo’s Dialogue concerning the two chief world systems. Counting trees. To see the relation between Euler’s problem of counting the number of ways to triangulate an .n 2/-gon, Catalan’s problem of par- enthetization and the count of planar,C rooted, trivalent trees, consider the diagram

3 3 3 3 3 4 2 4 2 4 2 4 2 4 2

5 1 5 1 5 1 5 1 5 1 1 3 45 1 2 4 5 1 3 5 1 2 4 5 1 2 3 5

((ab)c)d (a(bc))d (ab)(cd ) a((bc)d ) a(b(cd ))

The trivalent property of our trees may be interpreted for parenthetization as the two inputs and one output of a multiplication. To get a sense of how the Catalan numbers describe many other phenomena, see Richard Stanley’s book Catalan Numbers which contains Igor Pak’s paper on the history of the Catalan numbers (Appendix B). n q-analogues. The Gaussian polynomials, or q-binomial coefficients, were k introduced by Gauss in an 1811 paper, in order to study questions about the behavior of cyclotomic polynomials. This idea and the role of q-binomial coefficients in the theory of partitions and number theory will be considered in other exercises. Analogy can be a powerful source of potential results, and here I study the Gaussian polynomials as if I were studying the binomial coefficients. The use of the lexicographic order for induction is justified because the initial conditions in each portion of the total order on pairs hold Style notes 253

m m m from the fact that 1 and that we can take 0 0 D D m m k D for k > 0. Wonderful  sources for properties of Gaussian polynomials C  are the books of George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge University Press, New York, 2004, and George E. Andrews, The Theory of Partitions, Cambridge University Press, New York, 1998. A breakthrough. These “notes” constitute a paraphrase, with mathematics, of a celebrated letter of Carl-Friedrich Gauss to Heinrich Olbers (1758–1840) dated September 3, 1805. Gauss refers to the work, which is recorded in his mathematical diary on August 30, 1805, that later appeared in his paper Summatio Quarumdam Serierum Singularium, Comm. soc. reg. sc. Göt- ting. rec. 1(1811). Of course, he would not have written so many mathe- matical details. In the actual letter to Olbers, he was particularly delighted in a result, and surprised by its discovery, an unusual example of self- congratulation. An account of the work and this letter appears in the essay by S.J. Patterson, Gauss sums, pp. 505–528 in the collection The Shap- ing of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, edited by Catherine Goldstein, Norbert Schappacher, and Joachim Schwermer, Springer-Verlag, Berlin Heidelberg, 2007. The polynomials, denoted by n Gauss as .n; k/, are the q-binomial coefficients, of the exercise q- k analogues.   Partitions of numbers. It was Leibniz in a letter of 1669 to Johann Bernoulli who posed the problem of counting the number of ways a given number can be written as a sum of parts. Euler developed the theory of partitions discovering many remarkable theorems, and establishing the approach via generating functions that is followed today. Among the gems that Euler found is his theorem relating partitions with even or odd numbers of sum- mands and the pentagonal numbers. Good places to read about these ideas are the books mentioned in the style note on q-analogues by Andrews. Ed- win’s maps of possible routes through Carré are called Ferrers diagrams, named for Norman M. Ferrers (1829–1903), an editor of the Cambridge and Dublin Mathematical Journal, and the Quarterly Journal of Pure and Applied Mathematics. Ferrers had communicated the idea of his diagrams to J.J. Sylvester (1814–1897) who made the use of Ferrers diagrams a sig- nificant tool to understand partitions. Take it to the limit. In this exercise I reverse history in order to relate the Gauss polynomials to the theory of partitions. By manipulating the restrictions on partitions we arrive at Euler’s product that gives the numbers p.N / as coefficients of the formal power series associated to the product. Euler’s famous book Introductio analysin infinitorium, Lausanne, 1748, contains the beginnings of partition theory (§16). Another engaging account of these 254 Style notes

ideas can be found in Lecture 3: Collecting like terms and missed oppor- tunities in Mathematical Omnibus by Dmitry Fuchs and Sergei Tabach- nikov, Amer. Math. Soc., Providence, RI, 2007. See also the book of Bruce C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, RI, 2006, who explores contributions of Srinivasa Ramanujan (1887–1920) to partition theory. q-binomial theorem. The q-binomial theorem was known around 1808 by Gauss, but he did not publish it. H.A. Rothe (1773–1841) called attention to the theorem in the preface of his book Systematisches Lehrbuch der Arithmetik, but he did not give a proof of it in the book. Among Gauss’s unpublished pa- pers, the q-binomial theorem is found in Hundert Theoreme über die neuen Transcendenten. His proof was by induction. These historical details can be found in the magisterial book of Ranjan Roy, Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century, Cambridge University Press, New York, 2011. The quantum plane. This form of the binomial theorem first appeared in a paper of Marcel-Paul Schützenberger, Une interprétation de certaines solutions de l’équation fonctionnelle: F .x y/ F .x/F .y/, C. R. Acad. Sci. Paris, 236(1953), 352–353. The studyC of noncommutingD quantities earned the ad- jective quantum in the late twentieth century. Quantum algebra became a fo- cus of much research thanks to the connection between Yang-Baxter equa- tions (physics) and knot theory. A clear introduction to this subject can be found in the book of Christian Kassel, Quantum Groups, Springer-Verlag, Heidelberg, 1995. Scrapsheet. The identity proved here is number 3.118 in the incredible collec- tion, Combinatorial Identities, compiled by Henry W. Gould, Morgantown, W.Va., 1972. This publication is subtitled a standardized set of tables listing 500 binomial coefficient . Reading it is like visiting a rich mine where everywhere you turn there seems to be gold, but you have to dig it out yourself. In this exercise, I tried to follow some unlikely proof ideas until they fail, as if I had just transcribed a scrapsheet of work in progress. Is a mathematician someone who turns coffee into theorems? The trick of multiplying by 1 where 1 has a helpful representation is used. One cannot underestimate the potential of this trick when ingeniously used. The coffee stains were designed by Freepik.com. Mathematical Idol. Which proof do you prefer? Pitting induction against the algebra of polynomial expressions was a lovely suggestion from my editors. xn yn In Hana’s proof, an analogy plays the key role, namely, between x y .x/n .y/n and . In his book Combinatorial Identities, Henry W. Gould x y Style notes 255

also compares the expressions considered in the exercise to

n x 1 1 y C k : x 1 n y k 1 k  C   XD Gould reports that the identity is known in Harry Bateman’s unpublished Notes on Binomial Coefficients as the Capelli relation. The graphic art for these pages was done by Anastasia Stevens. Symmetries. The symmetric polynomials play an important role in Galois the- ory, the study of polynomials and their roots. The computations here show how generating functions and the binomial theorem lead to expressions for xn yn in terms of x y and xy. This material is classical. A polynomial C C f .x1; x2; : : : ; xn/ is symmetric if for any permutation  of 1; 2; : : : ; n , f g f .x1; x2; : : : ; xn/ f .x.1/; x.2/; : : : ; x.n//. The fundamental theorem of symmetric polynomialsD states that such a polynomial may be expressed as a polynomial in the polynomials s1 x1 x2 xn, s2 D C C    C D xi xj , s3 xi xj xk,..., sn x1x2 xn. For a 1 i

and displays. The snake oil method of Wilf aims to establish an identity based on sums that depend on a parameter (in this case j ). Wilf suggests that we construct a generating function in two variables for the function of the parameter given by the sum. The crucial step (the snake oil) is the in- terchange of the order of summation after which simplifications are possi- ble, leading to identities. Many illustrative examples are presented in Wilf’s book generatingfunctionology. Rings and ideals. The definition of a commutative is sufficient for a proof of the binomial theorem for nonnegative integer exponents. (Can you prove it?) Rings and their ideals are often discussed in the first undergraduate course on abstract algebra. If you find these ideas unfamiliar, have a look in the book of Joseph Gallian, Contemporary Abstract Algebra, Cengage Learning, Boston, MA, 9th edition, 2017. The appearance of the binomial theorem in the proof that pI is an ideal is an unexpected trick worth know- ing. Ideals not only generalize the notion of a normal subgroup, but they play a key role in a more general notion of divisibility introduced by Kum- mer and Richard Dedekind (1831–1916) and developed by David Hilbert and Emmy Noether (1882–1935). The properties of ideals in the polyno- mial ring in several variables provide an algebraic foundation for algebraic geometry. The radical of an ideal in such a ring plays a key role in the proof of Hilbert’s Nullstellensatz. I refer the reader to Advanced Modern Algebra by Joseph Rotman, second edition, American Mathematical Society, Provi- dence, RI, 2010, for more details. Mathmärchen. The prose of the Grimm brothers’ fairy tales has always fasci- nated me. The tale of Herr Fix und Fertig (from the 1812/1815 collections) is the model for my tale. In the Grimm story the protagonist Mister Fix-it is kind to various animals (Frosch, frog; corbeau, crow; oshidori, duck) who help him win the hand of a princess in unexpected ways. The argument in the exercise appears in Peter J. Cameron’s lovely book Combinatorics: Top- ics, Techniques, Algorithms, Cambridge University Press, New York, 1994. The more usual proof of the existence of the fields of cardinality pn uses Galois’ theory of fields and the fact that xp x .mod p/ for all x in Z=pZ (Fermat’s little theorem). Methods of counting in finite algebraic structures can lead to unexpected places. The quote about generating functions and clothes lines is from Herb Wilf’s generatingfunctionology. For a thorough introduction to the Möbius function, see Chapter 2 of H.E. Rose, A Course in Number Theory, Oxford Univ. Press, NY, 2nd edition, 1994. The binomial distribution. Games of chance led to the analysis of repeated trials—flipping a coin or rolling a die give independent trials. Early ac- counts of the probability of k successes in n trials can be found in the work of Pascal and de Moivre. It was Jacob Bernoulli in his treatise Ars Conjectandi who formalized the binomial distribution. The lure of a one- Style notes 257

sentence exercise was irresistible, especially after reading the novel of Bo- humil Hrabal, Dancing Lessons for the Advanced in Age (1964), which is a single sentence of 117 pages. Ed Park’s review, One sentence tells it all (New York Times, Sunday Book Review, December 24, 2010) discusses other efforts at the “Very Long Sentence.” Wormhole points. One of the first discussions of probability theory was an ex- change of letters in 1654 between Pierre de Fermat and Blaise Pascal on the Problem of Points. In their discussion, two equally skilled players are inter- rupted while playing a game of chance for a pot of money. The score at that moment does not decide the winner. Based on the score and the game, how should the pot of money be divided? A translation of the exchange between Fermat and Pascal along with other historical background on probability and statistics can be found at ✇✇✇✳②♦❦✳❛❝✳✉❦✴❞❡♣✴♠❛❤✴❤✐❛✴♣❛❝❛❧✳♣❞❢✳

According to Victor Katz, in A History of Mathematics, HarperCollins Col- lege Publishers, New York, 1993, the problem of points played a motivating role in Pascal’s writing of the Traité du triangle arithmétique. You can’t always get what you want. The distribution that motivates this com- putation arises in the problem known as Banach’s matchbox problem, para- phrased in the exercise Matt Hu, graduate student. The probability in that 2n k 1 problem takes the form P rŒX k . To know that we 2n k D D n k 2 have a probability distribution we need to show 

n n 2n k 1 n .n k/ 1 C n k 22n k D n k 2n n k k 0   k 0   C XD XD n n m 1 C D m 2n m m 0 C XD   2n p 1 1: n 2p D p n D XD   The verse is based on the well-known Rolling Stones song You can’t always get what you want, written by Luigi Creatore, Keith Richards, Mick Jag- ger, Hugo Peretti, and George Weiss; Copyright: Abkco Music Inc., Gladys Music, Mirage Music Int. Ltd. c/o Essex Music Int. Ltd. I learned how the refrain of this song describes mathematical research from Haynes Miller. Matt Hu, graduate student. The problem of the distribution of the remaining number of candies is a reformulation of Banach’s matchbox problem that 258 Style notes

trades candies for matches (see William Feller, An Introduction to Prob- ability Theory and Its Applications, vol.1, 2nd edition, John Wiley and sons, New York, 1957, p. 157). The problem was probably introduced by Hugo Steinhaus and not by Stefan Banach. Banach was a heavy smoker. Fans of Raymond Chandler will recognize the first sentence as a para- phrase of the beginning of one of his novels. Stirling’s formula, that nŠ 1=2 n .1=2/ n  .2/ n C e , is an extraordinary result, established in 1730. A first approximation was made by Abraham de Moivre around the same time. Proofs of the formula abound on the web, and center around the value of log.nŠ/. The setting of the professor’s office recalls my advisor’s office. Seeking successes. The probability distribution P rŒk failures, n successes is known as the negative binomial distribution. It also allows us to compute the waiting time to the nth success for a repeated Bernoulli trial. The distri- bution was introduced by Pascal in letters to Fermat (see Varia Opera Math- ematica. D. Petri de Fermat. Tolosae (1679)). Pierre Rémond de Montmort used the negative binomial distribution to explore the number of times a fair coin need be tossed to obtain a certain number of heads. The readers of this sort of letter were thought to be technically savvy. TEXNO…AI€NIA. Technopaegnia is the term usually applied to classical Greek or Latin pattern poems that were written in a shape related to their content. The visual appearance of the poem added an extra dimension and additional force to the work. The form was revived in the Renaissance. Famous exam- ples include the poems “The Altar” and “Easter Wings” by George Her- bert (1593–1633). The form was again revived in the late 19th century as calligrammes by Mallarmé and Apollonaire, becoming in the 20th century concrete poetry to the Dadaists and later poets. Of course, a concrete poem concerning binomial coefficients must take the shape of a triangle. Each line has one more character than the last—punctuation doesn’t count. Each character in a mathematical expression is important. Calculus as a tool in the study of polynomials can never be underestimated. This proof and its gener- alizations are discussed in Combinatorial Enumeration by I.P. Goulden and D.M. Jackson, Wiley and Sons, New York, 1983 (also Dover Publications, Mineola, NY, 2004). An area computation. This computation of area was suggested by Pascal in his treatise Potestatum numericarum summa of 1654—see Oeuvres de Pascal, edited by Brunshvicg, Boutroux, and Gazier, 1909–1914, Paris; volume 3, pp. 346–367. Another version appears in Wallis’s Arithmetica infinitorum 1 1 of 1655. The fact that xk dx was known in the years before 0 D k 1 Newton and Leibniz introducedZ calculus.C The role of arithmetic in resolving geometric problems became apparent around this time, and soon blossoms into calculus. Style notes 259

Experimental mathematics. This fictitious letter is written in the manner of Isaac Newton (1643–1727), who explains to “Sir” that he has discovered patterns among the binomial coefficients that can be extended by interpolation to negative and to fractional exponents. Furthermore, together with term-by- term integration, these series solve geometric problems in an elegant and efficient manner. The entries in the tables are from Whiteside’s The Math- ematical Papers of Isaac Newton, volume 1, Cambridge University Press, Cambridge, 1967. The table is adapted from the frontispiece of volume 1, a sheet of calculations in Newton’s hand. These youthful computations re- quire rigorous proof, which Newton did not provide, though he verified them by algebraic means. His investigations of calculus began with the bi- nomial series. For more about Newton and the binomial theorem, see Chap- ter VII of William Dunham in Journey through Genius, Wiley and Sons, New York, 1990; D. T. Whiteside, Newton’s discovery of the general bi- nomial theorem, Math. Gazette, 45(1961), 175–180; and Niccolò Guiccia- rdini, Isaac Newton on Mathematical Certainty and Method, MIT Press, Cambridge, MA, 2011. In the printing of the time, Newton would write ye for “the,” yt for “that,” wch for “which,” and & for “and.” Telescopes. Gottfried Wilhelm Leibniz (1646 –1716) is acknowledged as one of the discoverers of calculus with Newton. He was famously a visitor in Paris and later in London where he made an impression on the local intellectual communities. The summing of the reciprocals of the triangular numbers via a telescoping sum is attributed to Leibniz, and this fictitious letter is built on his accomplishments. A discussion of Leibniz’s early mathematical ac- complishments may be found in the books Leibniz in Paris 1672–1676: His Growth to Mathematical Maturity by J. H. Hofmann, Cambridge Univer- sity Press, Cambridge, 1974, and The Early Mathematical Manuscripts of Leibniz, translated from the Latin Texts published by Carl Immanuel Ger- hardt with critical and historical notes by J. M. Child, Open Court, Chicago, 1920. There is a proof of the fundamental theorem of the calculus, based on the , that employs a telescoping sum. The famous note Historia et Origo Calculi Differentialis of Leibniz, written between 1714 and 1716, in which he described his contributions to the development of calculus, has an odd tone—Leibniz refers to himself in the third person. My fictitious letter seems in keeping with his manner of communication. The signature is, in fact, an alias Leibniz used to publish a politically sensitive treatise on the law. Historia et Origo also contains the harmonic triangle. The Rechner. In the days before computers, lengthy and tedious calculations were given to individuals or groups to carry out, freeing the principal inves- tigator for loftier pursuits. The term computer was coined for such folks, ein Rechner in German, and the term human computer is used now. D. A. Grier’s book When Computers Were Human (Princeton University Press, 2005) 260 Style notes

tells the story of the role of human computers in determining the orbit of Halley’s comet in 1758 and its later returns. I imagined my Rechenknecht as part of this effort. The presentation of the general binomial theorem for rational coefficients

m m m m n m 2n m 3n .P PQ/ n P n AQ BQ CQ DQ C D C n C 2n C 3n C 4n C   appears in Newton’s Epistola prior of 1676. Its effectiveness as an algo- rithm for computing roots is the point of this exercise. Lipogram. A lipogram is a constraint on a written piece that excludes certain chosen letters of the alphabet from the piece. I have excluded the letter e from the piece, as George Perec does in La Disparition. The story of the ex- ercise is based on the 1683 study of Jacob Bernoulli on compound interest, in which he shows that continuous compounding is bounded (Quèstiones nonnullae de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685 Acta eruditorum, 219–23). A key role is played by the binomial coefficients and limiting expressions for them. 18th century machinations. In this exercise I have followed Vandermonde’s Mé- moire sur des irrationnelles de différents ordres avec une application au cercle, Histoire de l’Acadèmie Royale des Sciences avec les Mèmoires de Mathèmatique et de Physique pour l’annèe 1772 (1772), 489–498, using modern notation. However, the reasoning is very 18th century in style. Sug- gestions of form lead to jumps in method, without a lot of discussion. For example, going from .x/n .x/0.x 0/n to .x/0 .x/n.x n/ n to the D D form for .y/ n, or more dramatically, the leap in 3 from .x k l/l .x/ l to an infinite series when k and l are rational. Without convergenceC C re- sults 18th century mathematics moved forward based on trust in algebra. My thanks to Jim Tattersall for a copy of his article on Vandermonde and a copy of the Mémoire. Wallis’s celebrated product formula for =2 was my goal of the exercise. Vandermonde also derived a formula of Newton for =2 in this manner from the . Term-by-term integration only requires the binomial theorem as input to achieve these results. Recipe. This exercise gives a sort of explicit recipe for the use of induction. I feel that one ought to use induction sparingly (like cardamon or capers) because it usually comes after a pattern is found. Finding the pattern often leads to a proof that keeps closer to the objects at hand. This proof applies term-by- term differentiation. If a function f .x/ is represented by an infinite series, 1 k say, f .x/ ak.x x0/ for x x0 < r, then at any point x1 inside k 0 D D j j the interval ofX convergence .x0 r; x0 r/, term-by-term differentiation and integration are allowed. See the classicC text of Konrad Knopp, Theory and Application of Infinite Series, Dover Publications, Mineola, NY, 1990, Style notes 261

for a proof. The key condition is uniform convergence of series. Differenti- ation can be useful for establishing identities between series associated with binomials. Don’t get me wrong—induction shouldn’t be avoided! The right tool for the job makes work easier. Math talk. Fans of National Public Radio’s Car Talk will certainly recognize this kind of banter. The names Jack and Jake refer to Johann (1667–1748) and Jacob (1655–1705) Bernoulli, brothers but also rivals in mathematics. The d operator x applied to a power series in x has the effect of introducing a dx factor of the exponent at each term:

d 1 k 1 k x akx kakx : dx D k 0 k 0 XD XD And iterating the operator accumulates factors of k. This formal trick leads to relations between generating functions and explains how powers can arise as coefficients. Averages and estimates. Average values and estimates play a role in some of the other exercises. They are also a tool in probabilistic methods in combina- torics. For examples, see Jiri Matoušek, and Jan Vondrák,J., The Probabilis- tic Method, lecture notes that can be found on the web, and the important book by Noga Alon and Joel H. Spencer, The Probabilistic Method, fourth edition, Wiley, New York, 2016. Average values and estimates of numerical functions also play a role in number theory. To get better estimates of the binomial coefficients, use Stirling’s approximation for nŠ, n n nŠ p2n :  e   Equality. The opening question, “What does .1 x/p2 mean?”, is something that is not always asked in elementary calculusC classes. The proposal to represent such a function of x as a power series leads to other questions— what does it mean to represent a function by a power series? What is the radius of convergence? How does such a function depend on the exponent? I based my answers to these questions on Konrad Knopp’s exposition of similar issues in the 1928 classic, Theory and Application of Infinite Series, available from Dover Publications, Minelola, NY, 1990. If we take Euler’s approach and let .1 x/r er ln.1 x/, then we get a series representation C D C p2 p2 ln.1 x/ .1 x/ e C C D 2 ln2.1 x/ .p2/3 ln3.1 x/ 1 ln.1 x/p2 C C : D C C C 2Š C 3Š C    262 Style notes

k s.k; j / Comparing this with .1 x/r 1 rj xk we can ob- k 0 j 0 kŠ C Dn D D tain a series representations for ln X.1 x/X. It is possible to generalize the argument in the exercise to complex exponentsC and a complex variable x. , by parts. Calculation of  has fascinated mathematicians since antiquity. This calculation of  in Newtonian terms is outlined in Newton’s notes on Wal- lis’s Arithmetica Infinitorum. (See D. T. Whiteside, Newton’s discovery of the general binomial theorem, Math. Gazette, 45(1961), 175–180.) His bet- ter known computation uses the same sort of reasoning with the circle of radius 1 centered at x 1. The series of Euler was obtained around 1755 via a series developmentD of the arctan function. The beta functions appear in Euler’s work on the gamma function, which is his interpolation of the func- tion on positive integers n nŠ. The integral representation of the numbers B.k; l/ offer many possible7! avenues of study, including and summations. For more details about Euler’s integrals, see the Chau- venet prize winning essay of Philip J. Davis, Leonhard Euler’s Integral: a historical profile of the Gamma function, Amer. Math. Monthly 66(1959), 849–869. Integration by parts appeared first in work of Pascal (1658) re- lating volumes and what would later become integrals. The method was known to Newton and Leibniz in their development of calculus. A macaronic sonnet. Macaronic poetry refers to verse that uses several languages at once. It is found sometimes in choral music—for example, In Dulci Ju- bilo, the Christmas carol. Is mathematics a universal language? Some be- lieve so. And consequently, no matter the language, mathematical ideas will get understood. This premise is tested in this poem. Though “unnecessary,” I provide a translation: Definitions offer unexpected delights to us all, From the low-hanging fruit that two ideas might make fall. An example, constructed from the simplest of series,

1 1 1 t t 2 t k; 1 t D C C C    D k 1 XD Emerges from a long look at the beta function:

1 k 1 l B.k; l 1/ t .1 t/ dt; defined for k 1; l 0. C D   Z0 What would happen if we multiplied the by .1 x/l , Then integrated over the unit interval?

1 1 1 l 1 1 l .1 t/ dt .1 t/ dt l D D 1 t Z0 Z0 Style notes 263

1 1 k 1 l t .1 t/ dt D 0 Z k 1 ! XD 1 1 k 1 l 1 t .1 t/ dt B.k; l 1/: D 0 D C k 1 Z k 1 XD XD But integration by parts allows us to immediately prove that

l B.k; l 1/ B.k 1; l/ C D k C l l 1 l .l 1/ B.k l; 1/ D    D k k 1    k l 1 C C C 1 : D k l k C l  

1 1 1 Thus we find, for n 1, that .  k l D l k 1 k C XD l   A hidden integral. This identity is numbered 4.9 in Gould’s Combinatorial Iden- tities among the sums of type 1=1. The sum invites simplifications of the ex- pression, but the value was not evident to me from such maneuvers. I some- times suggest to my students that there are often only so many things we can draw on; coaxing one of them to lead to a desired result, like 1 k=k, is as satisfying as devising a new approach. D Letter to a princess. The first successful popularization of science by a scientist was Leonhard Euler’s Letters to a German Princess on Subjects in Physics and Philosophy, a collection of 234 letters written by Euler in French to Friederike Charlotte of Brandenburg-Schwedt and her sister Louise between 1760 and 1762. The collected letters appeared in print between 1768 and 1774 in three volumes, supported by empress Catherine II. A translation into English was made by Henry Hunter in 1795. The subjects treated by Euler in the letters concerned physical phenomena for the most part: sound, air, temperature, light, optics, gravity, and astronomy. The letters explained the natural philosophy of the day. They were very popular in their time and enjoyed several editions. This exercise is based on the style of Euler’s let- ters, but it treats his mathematics, in particular his investigation of infinite series of the form 1 2k 3k 4k . The polynomials introduced here are known as Eulerian polynomialsC C and   their coefficients the Eulerian num- bers. More of their properties are developed in other exercises. The book of T. Kyle Petersen, Eulerian Numbers, Birkhäuer, New York, 2015, treats 264 Style notes

the basics and some of the more subtle combinatorial appearances of these numbers. Explicit Eulerian numbers. This exercise follows the example of the exercise n An explicit formula in which a formula for the binomial coefficient is k given only in terms of n and k. The symmetry of the binomial coefficients  can be seen in the explicit formula. The increasing property that leads to unimodality is proved in the exercise Up and down. These properties for n the Eulerian numbers are established in this exercise by a comparison k of generating functionsD andE induction on n. Worpitzky’s identity first ap- peared in the paper of J. Worpitzky Studien über die Bernoullischen und Eulerischen Zahlen, J. reine angew. Math. 94(1883), 203–232. Mutperation satticsits. Permutation statistics associate a number to each per- mutation of n letters. In the exercise we count rises, an artifact of the one line presentation of a permutation. This number gives a filtering of the per- mutations into subsets by number of rises. This statistic appears to have been introduced by Percy A. MacMahon (1854–1929) in his paper Second memoir on the composition of numbers, Phil. Trans. Royal Soc. London, A, 207(1908), 65–134. The result was an application of his Master Theo- rem. The route from compositions of numbers—partitions of numbers into ordered pieces—to counting rises is how MacMahon discovered the con- nection between Eulerian numbers and permutations. The words have been permuted in this exercise, following the example of Queneau exercise, Per- mutations par groupes croissants de lettres and Permutations par groupes croissants de mots. It is surprisingly easy to unscramble the words as you read, which tells us something about how the brain wants to know what is going on. Review. George Boole (1815–1864) is best known for his work on logic. How- ever, his books A Treatise on Differential Equations, MacMillan and Co., Cambridge, 1859, and the book reviewed in the exercise became standard textbooks in the late 19th century in England. Both are in print today as classic texts available from Cambridge University Press. The theory of fi- nite differences continues to have a place in mathematics, especially in nu- x merical methods and dynamical systems. The difference equation  n D x makes the binomial coefficients particularly interesting. Several  of n 1 the identities found in this book may be proved from a finite difference point of view. Furthermore, interpolations play a role in Newton’s and Leibniz’s discoveries of calculus. Plus C . This exercise follows the argument in Abel’s original paper, Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist. Crelle J. 1(1826), 159–160, with the addition of modern summation nota- Style notes 265

tion. Integration is a natural tool for induction involving polynomials be- cause it increases degree. Of course, you have to determine the . Alternatively, Abel’s identity may be proved by induction and the use of the method of undetermined coefficients. Such a proof is given in Eugen Netto’s book, Lehrbuch der Combinatorik, Teubner, Leipzig, 1901. There are several generalizations of Abel’s generalization of the binomial theorem, including one by Adolf Hurwitz, Über Abel’s Verallgemeinerung der binomischen Formel, Acta Math., 26(1902), 199–203. A survey of these kinds of theorems may be found in the paper of A. Kelmansa, and A. Post- nikov, Generalizations of Abel’s and Hurwitz’s identities, European J. of Combinatorics, 29(2008), 1535–1543. Googling “Abel’s binomial theorem” also obtains the form n 1 n n n k 1 k w .z w n/ .w n k/ .z k/ : C C D k C C k 0 XD   At a carnival. Every a ib is a binomial and so when powers are taken, the binomial theorem applies.C The formulas presented here are due to Abraham de Moivre (1667–1754) around 1722, and they were proved in detail by Euler in 1749. The exercise presents something paradigmatic of mathematical development: New viewpoints (complex numbers) simplify formulas (trig identities) that appeared complicated. Inspiration for the style of this exercise came from the Tom Waits’s song Step right up on his album Small Change. Indicators. It is a natural question to ask for the sum of a periodic subset (for example, every third element) of a family of binomial coefficients. I don’t believe the term indicator is standard, and something like character is more usual. However, I wanted to emphasize the key property of an indicator: it is nonzero when a desired property holds, and is zero when the property does not hold. This exercise is based on the article by David Guichard, Sums of selected binomial coefficients, Math. Mag., 26(1995) 209–214. There is also an account of these ideas in Eugen Netto’s book, Lehrbuch der Com- binatorik, Teubner, Leipzig, 1901. The reader who wants to explore further n might work out the sum where 0 < r < m. r mj r n mj r  C   C  Tweets. Tweets (postings to Twitter.com)X are restricted to at most 140 charac- ters (including spaces and punctuation). It is a challenge to communicate in such an abbreviated form of discourse. This exchange is based on the well written paper of Henry W. Gould, A baker’s dozen proofs of a binomial identity, Univ. Beograd. Publ. Elek. Fak., 16(2005), 131–138. The identity n x x 1 tweeted here is . 1/ . 1/n and it appears as iden- k 0 k D D n tity 1.5 in Gould’sX book. I suggest  working out the details of the proofs—by 266 Style notes

a telescoping sum (Leibniz?), induction (Fermat?), comparison of polyno- mials (Kronecker?), a special case of the Chu-Vandermonde identity, and by a complex integral (Cauchy?). Then read Gould’s paper for even more approaches to proving the identity. Gould’s paper illustrates nicely how in- formative different approaches to the same problem can be. Beautiful numbers. I learned this derivation from the magisterial volume of Isaac J. Schwatt, An Introduction to the Operations with Series, The Press of the University of Pennsylvania, Philadelphia, 1924. The point of this exercise is not really the beauty of the formula, which is certainly open to discussion. The derivation is noteworthy because so many numerical domains are vis- ited, in which new simplifications are found. The beauty of Euler’s formula lies in why it is true. The same is true for Schwatt’s identity (1.90 in Gould). Reminiscences. This exercise is based on my memories of Emil Grosswald (1912– 1989). To add one more, I had a funny, unsuccessful idea about Fermat’s last theorem that I eventually developed and wrote up. I sent a first draft of my ideas to Grosswald, and I was pleased to receive a letter with detailed corrections from him. He improved the work considerably. The Egorychev method, sometimes called the method of coefficients, has its roots in the use of generating functions and the theory of residues. Egorychev’s book ap- peared in 1977 in Russian, appearing in 1984 in the Amer. Math. Soc. Trans- lations series. His proof of Grosswald’s identity appears on page 28. Many other interesting identities can be found in Egorychev’s book. Another pre- sentation made be found in the paper The method of coefficieints, by D. Mer- lini, R. Sprugnoli, and M.C. Verri, Amer. Math. Monthly 114(2007), 40–57. Winter journa.l The story Le Voyage d’hiver on which this exercise is based was written by Georges Perec in 1979. Since then other members of the OuLiPo have contributed an extension of Perec’s story collected in the book Win- ter Journeys by Perec and the OuLiPo, Atlas Press, London, 2013. Perec’s short story is developed by twenty other writers. Mathematical writing en- joys this sort of potential and a publication is most prized when it spawns many new ideas. The mathematical relation discussed is one of the relations of Moriarty type, as named by Harold Thayer Davis in The Summation of Series, Principia Press, San Antonio, 1962 (now available from Dover Pub- lications, Mineola, NY, 2015). There are also papers of Henry W. Gould on this topic that offer other proofs of the identities of Moriarty type and more Holmesiana. See The case of the strange binomial identities of Professor Moriarty, Fibonacci Quart., 10(1972), 381-391, and The design of the four binomial identities: Moriarty intervenes, Fibonacci Quart., 12(1974), 300- 308. Cellular automata. Cellular automata are determined by deterministic and sim- ple rules that sometimes lead to complex outcomes. The most well known example is the planar cellular automaton Life introduced by John Conway. Style notes 267

(See Martin Gardner, Mathematical Games: The fantastic of John Conway’s new solitaire game “life,” Scientific American, 223(1970), 120–123.) Using the Pascal identity (mod 2) as a generating rule leads to a relation between the Pascal triangle with the Sierpinski´ triangle. When pre- sented as a symmetric triangle, the Pascal triangle (mod 2) takes the form on the left and the usual portrait of the Sierpinski´ triangle is the array of triangles on the right.

The study of cellular automata has been championed by Stephen Wolfram. More details about the relation between cellular automata and the binomial coefficients may be found in the articles of Wolfram and Andrew Granville listed in the footnote. For more about the world of cellular automata, see Wolfram’s book A New Kind of Science, Champaign, IL, Wolfram Media, 2002. Tiling. The study of tilings of the plane has its origins in earliest times, espe- cially in the work of artists. From the tessellations of ancient Rome to the works of the twentieth century artist M. C. Escher (1898–1972) the repe- tition of patterns can be found in most cultures. The mathematics of tiling emerged with the language of symmetry, and developed in many directions when aperiodic tilings were discovered. One can read deeply about tilings in the best known book, Branko Grünbaum, G.C. Shephard, Tilings and Patterns, W.H. Freeman and Co., New York, 1987 (second edition, Dover Publications, Mineola, NY, 2016). A more recent account may be found in Charles Radin, Miles of Tiles, Student Math. Lib., vol. 1, Amer. Math. Soc., Providence, RI, 1999. The fractal aspects of the Pascal triangle tiling are dis- cussed in chapter 8 of the book by H.-O. Peitgen, H. Jürgens, and S. Saupe, Chaos and Fractals, Springer-Verlag, New York, 1992. More advanced the- ory has been surveyed by Natalie P. Frank in A primer of substitution tilings of the Euclidean plane, Expo. Math. 26(2008), 295–326. Lattice points. The geometry of numbers, developed by Hermann Minkowski (1864–1909), relates the counting of lattice points to problems in number theory. Minkowski’s theorem provides a relation between volume of a sym- metric convex body and the number of lattice points in that body. The poly- nomial associated with dilations of a polyhedron were developed by Eu- 268 Style notes

gène Ehrhart in many papers beginning with Sur les polyhèdres rationnels homothétiques à n dimensions, C. R. Acad. Sci. Paris 254(1962), 616–618, and his book Polynômes arithmétiques et méthode des polyhèdres en com- binatoire, Birkhäuser Verlag, Basel, 1977. Ian G. MacDonald proved the combinatorial reciprocity of the Ehrhart polynomials in Polynomials asso- ciated with finite cell-complexes, J. London Math. Soc. 4(1971), 181–192. A clear and engaging account of these ideas may be found in the book of Matthias Beck and Sinai Robins, Computing the Continuous Discretely, Integer-point Enumeration in Polyhedra, UTM, Springer-Verlag, New York, second edition, 2015. Afterlife. Both Newton and Abel studied the binomial series deeply. The ques- tion of convergence emerged as the methods of calculus were developed in the 18th century. Abel’s 1826 paper, Untersuchungen über die Reihe:

m m .m 1/ m .m 1/ .m 2/ 1 x  x2   x3 usw., C 1 C 1 2  C 1 2 3  C       Crelle J. 1(1826), 311–339, gave the first rigorous proof of convergence (when x < 1) and, though flawed in the assumption of uniform conver- gence,j indicatedj what is needed in order to be certain of the use of infinite series when representing a function. Some of the lines in this dialogue are direct quotes or slight paraphrases from the writings of Newton and Abel. See the book of Richard Westfall, Never at Rest : a Biography of Isaac Newton, Cambridge University Press, New York, 1980, and the work of Henrik Kragh Sørensen, especially the paper Exceptions and counterexam- ples: understanding Abel’s comment on Cauchy’s Theorem, Historia Math- ematica 32(2005), 453–480. Many of the quotes from Abel are found in letters to his teacher and friend B. M. Holmboe (1795–1850). A change in style from mathematics as natural philosophy to pure mathematics certainly takes place in the period between the lives of the interlocutors, and consid- ering their contributions to the history of the binomial theorem points up this change. The afterlife setting of the dialogue is inspired by the wonder- ful book by David Eagleman, Sum: Forty Tales from the Afterlives (Vintage, New York, 2010). Matt Hu and the Euler caper. This exercise was inspired by the lovely article of James Tanton, A dozen questions about Pascal’s triangle, Math Hori- zons, 16(2008), 5–30. My proof uses the Euler characteristic as a tool to do the counting. Tanton has another proof of the final formula. The pa- per of M. Noy, A short solution of a problem in combinatorial geometry, Math Mag. 69(1996), 52–53, also establishes the formula from the Euler characteristic. The Euler characteristic plays an important role in combina- torics and in topology. For an introduction to many applications of the Euler characteristic, see the excellent book Euler’s Gem: The Polyhedron Formula Style notes 269

and the Birth of Topology by David Richeson, Princeton University Press, Princeton, NJ, 2008.

Hypergeometric musings. Euler introduced the hypergeometric series 2F1 as a solution to a second order . He developed transforma- tion rules among such series in his paper Specimen transfromationis sin- gularis serierum, Nova Acta Acad. Petropol. 7(1778) 58–70, also in Opera Omnia, series 2, 16, 41–55 (E710). Gauss extended these researches follow- ing some developments by his PhD advisor, Johann Friedrich Pfaff (1765– ˛ˇ 1825). His paper Disquisitiones Generales Circa Seriem Infinitam x .1 ˛.˛ 1/ˇ.ˇ 1/ ˛.˛ 1/.˛ 2/ˇ.ˇ 1/.ˇ 2/    C C x2 C C C C x3 etc. Pars C .1 . 1// C .1 . 1/. 2/ C Prior, Comm. C Soc. Regiae  Sci. Gottingensis C RecentioresC , Vol. II. 1812 (reprinted in Gesammelte Werke, Bd. 3, 123–163 and 207–229, 1866) con- tains his systematic development of hypergeometric series. The case of higher degree polynomials P.k/ and Q.k/, or generalized hypergeomet- ric functions, was introduced by Leo August Pochhammer in Hyperge- ometrische Functionen nter Ordnung, J. reine angew. Math. 71(1870) 316– 352. Pochhammer also introduced notation for the hypergeometric func- tions. The relations between binomial identities and hypergeometric se- ries has long been recognized. A nice introduction is the paper of Ranjan Roy, Binomial identities and hypergeometric series, Amer. Math. Monthly, 94(1987), 36–46. This connection may be made more formal, and it sup- plies an ingredient in the work of Wilf and Zeilberger on the verifiability of binomial identities. On the bus. The story told here will be familiar to anyone who has opened Ray- mond Queneau’s Execices de Style. The inspiration for the t-shirt is a photo on Doron Zeilberger’s website, showing him in just such a t-shirt together with the question “Who you gonna call?” The Pfaff-Saalschütz Theorem was known around 1797 by Pfaff and it was rediscovered by L. Saalschütz in 1890. A nice proof may be found in the paper of J. Dougall, On Van- dermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25(1907), 33–47. Passing the identity through a test for being a hypergeometric series is a simple case of the method of Sister Celine Fasenmeyer (1906–1996) introduced in her PhD thesis Some generalized hypergeometric polynomials, Bull. Amer. Math. Soc. 53 (1947), 806–812 (1945 University of Michigan PhD thesis). The method was extended by R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75(1978), 40–42, and by Herb Wilf and Doron Zeilberger, An Algorithmic Proof Theory for Hypergeometric (Ordinary and “q”) Multisum/Integral Identities. Invent. Math. 108(1992), 575–633. Their method is far-reaching and can be used to discover identities algorith- mically. A description of these developments can be found in the book of 270 Style notes

M. Petkovsek, H.S. Wilf, D. Zeilberger, A=B, A K Peters (1996). Shalosh mentioned in the exercise is Shalosh B. Ekhad, a distinguished colleague of Doron Zeilberger.

I came to depend on a few books that have not been mentioned because they treat the foundations of combinatorial thinking. I cannot fail to mention Louis Comtet, Advanced Combinatorics: The Art of Finite and Infinite Ex- pressions, D. Reidel Pub. Co. Boston, 1974, a translation by J.W. Neinhuys of Analyse Combinatoire, tomes I et II, Presses Universitaires de France, Paris, 1970; John Riordan, Introduction to Combinatorial Analysis, Dover Publications, Mineola, NY, 2002, an unabridged republication of the work published by John Wiley & Sons, Inc., New York, 1958; Ron L. Graham, Donald E. Knuth, and O. Patashnik, Concrete Mathemat- ics, Addison-Wesley Pub. Co., Reading, MA, 1990. I recommend these books for further exploration of combinatorial ideas. The quote from Ian Hacking in the preface is from his paper ‘Style’ for Historians and Philosophers. Studies in the History and Philosophy of Science, 23(1992), 1–20. The Seven-Percent Solution by Nicholas Meyer was published in 1974 by E.P. Dutton, New York. The mathematical papers of Raymond Queneau are Sur les suites s-additives, C. R. Acad. Sci. Paris Sér. A-B 266(1968) A957–A958, and Sur les suites s-additives, J. Combinatorial Theory Ser. A 12(1972), 31–71. Finally, for English readers, there is a superb translation (with some additions) of Queneau’s Exercices de Style, published as Exercises in Style, and translated by Barbara Wright and Chris Clarke, New Directions, New York, NY, 2013. Index

n , 1 , 100 k Cn, 104, 107 Tn, 11 ŒnŠ , 109 Pn, 11 q N n Œ , 39 , 109, 116, 118, 123 X k , 41   k .n; k/, 112 < Ln k 1;k, 41 pn;k.N /, 116, 118 ŒnC p .N /, 118 , 42 k 2 p.N /, 119 Ln;k , 42 FN .q; t/, 120 R.k; l/, 44 Bk, 136 Dn, 51 R=I , 140 MN , 56 pI , 141 n , 57 Fp, 142 a1; a2; : : : ; at n .n/, 144  , 59, 90  EŒX, 145, 153 k m m a b .mod p/, 63 P n , 166  n ep.n/, 65 Œp , 170 r , 65, 206 d b cn , 73 x , 177 k 0 dx D , 184, 185 HYn, 74 1 2 i .n 1/ n, 77 B.k; l/, 184, 186         n ai , 83 , 192, 192 i k x Y , 86b †D n,E 195 k f .x/, 197 p=q   , 88 Ef .x/, 198 k f .x/, 199 .x/ n, 93, 131, 170 s.n; k/, 94, 181 !Xm, 206 S.n; k/, 94, 101 z, 207 n b , 213 P , 96 1 D k n, 224   S, 97   L.P; N /, 226 M 0, 97 V E F , 231 C SUR.Œn; Œk/, 100 a1; a2; : : : ; ap pFq z , 234 HOM.Œn; Œk/, 100 b1; b2; : : : ; bq I   271 272 Index

Abel, Niels Henrik (1802–1829), 200, 227, Chebyshev polynomials, 205, 216 250, 264, 268 choices with repetition, 37, 42, 102, 143 absorption identity, 8, 20, 59, 111, 146 Chu Shih-Chieh (13th century), 24, 27, 242 actuaries, 155 Chu-Vandermonde identity, 27, 87, 209 alchemy, 74, 248 closed form, 20, 61, 66, 106, 126, 157, 184, algebra, 5, 16, 27, 57, 86, 90, 126, 140, 192, 210 142, 170, 195 coffee, 126 algorithm, 79, 81 combinatorial reciprocity, 226, 268 alternating sum of powers, 189 , 140 approximation, 166, 179 complete graph, 43 area under a curve, 158 complex integrals, 209, 213, 266 arithmetical triangle, 1, 6, 24, 161 complex numbers, 204, 206, 210, 213 averages, 178 compound interest, 168 concrete poetry, 157, 258 Babbage, Charles (1791–1871), 79 conjugation, 207, 210 Banach, Stefan (1892–1945), 258 convergence, 88, 174, 180, 228, 261, 268 Banach’s matchbox problem, 152, 258 counting, 3, 7, 9, 18, 22, 25, 28, 31, 37, Bernoulli numbers, 136, 248, 255 39, 46, 51, 54, 59, 100, 107, 115, 118, Bernoulli, Jacob (1655–1705), 255, 256, 147, 195, 224, 232 260 Bernoulli, Johann (1667–1748), 253, 261 de Moivre, Abraham (1667–1754), 90, 216 Bertrand’s Postulate, 83, 249 decision tree, 4 beta function, 184, 186, 262 derangement number, 51 bijective proof, 18, 26, 41, 241, 244 derangement, 51, 245 binary counting, 23, 26, 242 derivative, 157, 174, 176, 190 binomial coefficients modulo a prime, 71, dialogue, 104, 175, 227 247 Dies Irae, 14, 240 binomial distribution, 145, 256 distribution problem, 31, 59 binomial inversion, 49, 52, 96, 100, 244, divergent series, 189, 228 251 divisibility, 62, 65, 75, 77, 81 Blom, L. 215 Doyle, Arthur Conan (1859–1930), 215 Boole, George (1815–1864), 197, 264 bounds, 179 Egorychev’s method, 214, 266 Bourbaki, 99, 251 Ehrhart polynomials, 225, 268 Erdos,˝ Pàl (1913–1996), 68, 83, 247, 249 calculus, 75, 157, 174, 176, 183, 197, 259, 268 estimates, 29, 83, 154, 179, 261 Carré, 9, 25, 28, 115, 239, 242 Euclid, 12, 240 carry, 67 Euler characteristic, 231, 268 Catalan numbers, 106, 108, 251 Euler integral of the first kind, 184, 187 Catalan, Eugene C. (1814–1894), 108, 252 Euler, Leonhard (1707–1783), 1, 108, 119, Cauchy, Auguste (1789–1857), 180, 213, 184, 189, 204, 234, 246, 250, 252, 250, 266 253, 262, 263, 265, 268 cellular automata, 218, 266 Euler’s formula, 210 central binomial coefficient, 77, 83, 178 Eulerian numbers, 192, 192, 196, 264 chair-committee identity, 7 expected value, 145, 153, 156 change-of-basis, 94 explicit formula, 20, 22, 52, 106, 135, 192 Index 273 exponential generating function, 135, 138, Grosswald, Emil (1912–1989), 212, 266 255 haiku, 62, 246 fairy tale, 142 Halley, Edmond (1656–1742), 166 falling , 93, 170, 198, 242 Hardy, G.H. (1877–1947), 247, 249 Fasenmeyer, Celine (1906–1996), 269 harmonic numbers, 74, 77 Fermat, Pierre de (1601–1665), 63, 246, harmonic triangle, 164, 259 257 hidden variable, 176 Fermat’s little theorem, 64, 71, 142, 246, higher parabolas, 158 256 hockey stick relation, 34, 39, 111 Ferrer diagrams, 115, 253 Holmes, Sherlock, vii, 215 Ferrers, Norman M. (1829–1903), 253 human computers, 166, 259 fibomomials, 243 Huygens, Christiaan (1629–1695), 163, Fibonacci sequence, 35, 243 215 figurative numbers, 11, 24, 240, 255 hypergeometric series, 233, 236, 269 finite differences, 167, 197, 264 finite field, 71, 142 ideal, 140, 256 finite integral, 199 indicator, 206, 265 Flatland, 240 induction, viii, 33, 35, 38, 45, 63, 83, 86, Fleming, Ian (1908–1962), 215 121, 124, 129, 150, 159, 168, 173, formula of Legendre, 66, 84 190, 193, 201, 209, 222, 231, 246, formula of Touchard, 56, 245 252, 260 formulas for , 162, 172, 184, 262 inequalities, 178 fundamental theorem, symmetric polyno- integer lattice, 224 mials, 134 integration by parts, 183, 186 interpolation, 161, 167, 197, 259, 266 games, 43, 244 inversion, seebinomialinversion Gauss identity, 234 irrational exponents, 180 Gauss sum, 112 irreducible polynomials, 142 Gauss, Carl Friedrich (1777–1855), 63, 71, 109, 112, 234, 246, 253, 254, 269 joke, 273 Gaussian polynomials, 109, 112, 118, 123, 253, 254 Kemeny, John G. (1926–1992), 97, 251 generalized binomial theorem, 88, 161, Kummer, Ernst (1810–1893), 67, 256 180, 249 Kummer’s theorem, 67, 68, 222, 247 generalized hypergeometric functions, 233, 236, 269 lattice paths, 9, 25, 28 generating function, 102, 104, 116, 120, Laurent series, 213 132, 135, 138, 143, 190, 251 LeBlanc, M. (Germain, Sophie) (1776– geometric representation, 12 1831), 112 geometric series, 88, 132, 173, 186, 190, Legendre, Adrien-Marie (1752–1833), 66 216 Leibniz, Gottfried (1646–1716), 163, 217, Glaisher’s theorem, 72 245, 253, 259 graduate school, 152 Leonardo of Pisa (ca. 1170– ca. 1250), 35, Granville, Andrew (1962–), 69, 82, 220, 243 247, 248, 249, 267 letter, 51, 112, 155, 161, 163, 189 graphic novel, 129 limerick, 29, 242 274 Index limits, 52, 118, 169, 180, 189 Pascal identity, 5, 7, 10, 15, 33, 37, 55, 86, linear algebra, 93, 96, 250 97, 218, 244 lipogram, viii, 168, 260 —for multinomials, 58 long division, 131 —for other numbers, 59, 90, 109, 113, 124, Lucas’s theorem, 72, 247 192 Pascal triangle, 1, 17, 21, 23, 30, 59, 68, macaronic poem, 186, 262 161, 223, 240, 250 MacMahon, Percy A. (1854–1929), 264 Pascal-de Moivre triangle, 59, 90, 250 Mann–Shanks theorem, 81, 249 Pascal, Blaise (1623–1662), vii, 24, 159, matrix derivative, 97 239, 257, 258 Matt Hu, 152, 209, 230, 257 Perec, Georges (1936–1982)), viii, 260, ménage numbers, 56, 245 266 method of coefficients, 214, 266 permutations, 51, 195, 255, 264 milk, 209 Pfaff-Saalschütz identity, 237, 269 Minkowski, Herrmann (1864–1909), 267 Pingala˙ (ca. 200 B.C.E.), 23, 242 modular arithmetic, 63, 71, 79, 247 Pochhammer, Leo A. (1841–1920), 269 Möbius inversion/function, 49, 144, 256 Pochhammer symbol, 93, 100, 109, 131, Moriarty identity, 216, 235, 266 170, 242, 254 Moriarty, James, vii, 215 polynomial identities, 86, 94, 103, 109, multinomial coefficients, 57, 60, 91, 245 112, 126, 129, 132, 170, 190, 200 multiplicative principle, 3, 7, 18, 22, 239, polynomial rings, 142, 256 241 polynomials in two variables, 132 music, 14 polytope, 224 powers of 11, 15 prime numbers, 77, 79, 81, 248, 249 n-simplex, 224 principle of inclusion and exclusion (PIE), negative binomial distribution, 155, 257 46, 55, 60, 244 Newton, Isaac (1643–1727), 15, 89, 161, probability distribution, 145, 150, 153, 166, 183, 227, 240, 259, 260, 262, 155, 256, 257 268 problème des ménages, 54, 245 Newton’s binomial theorem, 105, 166, 180, problem of points, 147, 257 250 proof by contradiction, 68, 247 Noether, Emmy (1882–1935), 256 proof without words, 11, 240 notation, 1, 33, 57, 63, 71, 73, 97, 116, 234, pseudocode, 79 239 number of poker hands, 22 q-binomial coefficients, 109, 116, 118, 120, 252, 254 Olbers, Heinrich W. (1758–1840), 112, q-Chu-Vandermonde identity, 122, 125 253 quadratic formula, 105, 252 one sentence, 145, 257 quantum binomial theorem, 123, 254 Oresme, Nicole (1320–1382), 74, 248 Queneau, Raymond (1903–1976), vii, 147, 244, 255, 269 p-adic representation, 65, 69, 72, 222, 247 partition of a set, 99 radical of an ideal, 141 partitions of numbers, 116, 118, 120, 253, Ramanujan, Srinivasa (1887–1920), 254 253, 264 Ramsey theory, 43, 244 parts of speech, 138, 255 rearrangement theorem, 181 Index 275 recipe, 173, 260 sums of powers, 135, 160, 189, 255 reciprocals of binomial coefficients, 163, surjection, 99 185, 186, 187 Sylvester, James Joseph (1814–1897), 253 recurrence relation, 33, 35, 37, 104, 108, symmetric expressions, 16, 18, 29, 132, 117, 137, 192, 195, 243 193, 255 reductio ad absurdum, 68, 247 repetitions, 37, 42, 102, 143, 244, 251 tanka, 62, 246 restricted distribution, 59, 116, 246 technopaegnia, 157, 258 restricted sums, 206 telescoping sum, 69, 159, 163, 209, 259 ring, 140 tetrahedral numbers, 11 rises, 195 theorem of Fine, 73 rising factorial, 233 theory of residues, 213, 266 Rolling Stones, 257 tiling the plane, 221, 267 roots of unity, 113, 206 topology, 231, 268 row sums, 16, 18, 26 trees, 4, 107, 252 rule of sum, 7, 239 triangular numbers, 11, 135, 159, 163, 240, 255, 259 Sanskrit prosody, 23, 241 trigonometry, 114, 203, 207, 211 Shalosh B. Ekhad, 237 Twitter, 209, 265 shift matrix, 97 shifted arithmetic triangle, 81 unimodal, 29, 193, 243, 264 Sierpinski´ triangle, 218, 267 slack variables, 225 Vandermonde, Alexandre-T. (1735–1796), snake oil method, 138, 255 27, 170, 242, 260 sonnet, 186 von Ettinghausen, Andreas (1796–1878), 1 squarefree numbers, 68, 247 Stirling numbers, 94, 101, 181, 250, 251 Wallis, John (1616–1703), 162, 172, 258, Stirling’s formula, 154, 258, 261 260 subset-of-a-subset identity, 19, 49, 127, Wilson’s theorem, 77, 248 241 Wolstenholme, Joseph (1829–1891), 77, substitution tilings, 221 248 summation notation, 5 Wolstenholme’s theorem, 77 John McCleary (Mathematical)

STYLE

EXERCISES IN

What does style mean in mathematics? Style is both how one does something and how one communicates what was done. In this book, the author investigates the worlds of the well-known numbers, the binomial coefficients. The author follows the example of Raymond Queneau's Exercises in Style, offeringThe the bookand reader celebratestheexploring 99joy storiesoffamiliar thewriting the Forjoyin rich collectionvarious anyofmathematics ematics,propertiesmathematics one styles. students interestedof from numbers. of by this high on in up. schoolmath-

For additional information and updates on this book, visit www.ams.org/bookpages/nml-50

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