Discrete Calculus: Methods for Counting

674 Pages·2016·8.791 MB·English

by  Carlo Mariconda, Alberto Tonolo (auth.)

UNITEXT 103
Carlo Mariconda · Alberto Tonolo
Discrete
Calculus
Methods for Counting
UNITEXT - La Matematica per il 3+2
Volume 103
Editor-in-chief
A. Quarteroni
Series editors
L. Ambrosio
P. Biscari
C. Ciliberto
M. Ledoux
W.J. Runggaldier
More information about this series at http://www.springer.com/series/5418
Carlo Mariconda Alberto Tonolo
(cid:129)
Discrete Calculus
Methods for Counting
123
CarloMariconda AlbertoTonolo
Dipartimento di Matematica Dipartimento di Matematica
Universitàdegli Studi diPadova Universitàdegli Studi diPadova
Padova Padova
Italy Italy
ISSN 2038-5722 ISSN 2038-5757 (electronic)
UNITEXT- LaMatematica peril 3+2
ISBN978-3-319-03037-1 ISBN978-3-319-03038-8 (eBook)
DOI 10.1007/978-3-319-03038-8
LibraryofCongressControlNumber:2016946021
©SpringerInternationalPublishingSwitzerland2016
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To Frank Sullivan
Preface
Several years ago, we were asked to give a second-year course for a Computer
Science Engineering degree at the University of Padova, with some mathematical
content about discrete mathematics: combinatorics, finite calculus, formal series,
approximation of finite sums, etc. We were unable at that time to find a suitable
book for our students: either the mathematical aspects were too profound and
abstract,orelse explanations werepoorand eachexercise was solvedbymeans of
some unexplainable tricks. That last aspect is particularly common in combina-
torics, where, too often, the basic results on the subject are so distant from their
applications that exercises often appear to have an “ad hoc” solution. Our first
purpose was to develop a method for explaining combinatorics to our students,
whichwouldallowthemtosolvecertainadvancedproblemswithsomefacility.We
achieved this goal after several attempts, ultimately meeting with the approval of
our students, the involuntary guinea pigs in our experiment, to whom we express
our thanks.
Inthepartofthebookdevotedtocombinatorics,roughlyathirdofthevolume,
we begin by relating every application to a very few essential basic mathematical
concepts. A key role in this is played by the sequences and collections, terms that
weprefertothemorewidelyused,butmoreambiguoustermsofarrangementsand
combinations: indeed, thinking of real life (combination locks), why should a
combination denote a non-ordered set of symbols? We spend some time on the
Basic Principles of combinatorics, like the Multiplication and Division Principles:
we strongly believe that passing over these subjects is the number one cause of
errors in the applications. We focus on the occupancy problems, where one pre-
scribes a fixed number of repetitions of the elements in a sequence or a collection,
and we thoroughly discuss the Inclusion/Exclusion Principle and its consequences
(derangements, partitions, etc.). In these first chapters, we encounter, of course,
some famous stars such as the factorials, binomials, derangements, and the Bell,
Catalan, Euler, and Stirling numbers: all of them are first defined via a combina-
torialcharacteristicpropertyandonlyafterwardexplicitlycomputed.Thisallowsus
vii
viii Preface
to prove most of their properties—like the recurrence ones—with some simple
combinatorial arguments instead of the more tedious inductive method.
Chapter6isdevotedtothetechniquesforcomputingthesumsofafinitenumber
of consecutive terms of a sequence. They reproduce and actually can be an intro-
ductiontothoseofthedifferentialcalculus:derivatives,primitives,thefundamental
theoremofcalculus,andeventheTaylorexpansionfindtheirdiscretecounterparts
here. We also encounter the harmonic numbers.
A substantial part of the book is devoted to power series and generating formal
series. In Chap. 7, we state the basic definitions and properties, though there are
somedelicatepointslikeclosedformsandcompositionsofformalpowerseriesthat
need more careful attention. The Basic Principle for occupancy problems explains
how the tools introduced here are useful to solve combinatorial problems.
Wemadeanefforttokeepseparateasmuchaspossiblethealgebraicproperties
and the convergence offormal power series, which is introduced and used only in
the subsequent Chap. 8. In this chapter, we compute the generating formal series
of the sequences of famous numbers such as those mentioned above, or the
FibonacciandtheBernoullinumbersintroducedhere.Asaby-product,weestimate
easily the Bernoulli numbers via their relation with the famous Riemann zeta
function.
Generatingformalseriesreturninthefollowingchapters(thoughthereadercan
skip thesewithoutlosing themaincontent),e.g., inChap.11concerningsymbolic
calculus,wheretheyplayaprominentroleinfindingthenumberofsequences,ofa
prescribedlengthandalphabet,thatdonotcontainagivenpattern,orincomputing
the odds in favor of the appearing of a pattern before another one.
Chapters 9 and 10 are devoted to recurrence relations. In the first of these, we
showhowtheserelationsarise.Asectionisdevotedtodiscretedynamicalsystems,
i.e., recurrence relations of the form xnþ1 ¼fðxnÞ. We give a thorough analysis
of the case where f is monotonic on an interval, and a simple proof of the famous
Sarkovskii’stheoremontheexistenceoforbitsofanyminimalperiod,oncethereis
apointofminimalperiod3:wedidnotfindthismaterialinothertexts.InChap.10,
wemainlydealwiththeclassicaltheoryofthelinearrecurrencerelations.Here,the
reader canfind analternativeresolutionmethodbasedongenerating formalseries,
which turns out also to be useful in the proofs of the main results. We expect that
the average reader will skip these, and therefore, they can be found in a separate
section. The chapter ends with the divide and conquer relations and the estimates
of the magnitude of their solutions, so useful in the analysis of algorithms.
Chapters 12 and 13 are devoted to the Euler–Maclaurin formula, which relates
the sum of the values of a smooth function f on the integers of an interval with its
integral onthesameintervalanditsconsequences like:theapproximation ofsums
andsumofseries,asymptoticestimatesforthepartialsumsofaseries,well-known
and unusual integral criteria for the convergence of a series, and integral conver-
gence, the trapezoidal methods, and the Hermite formula for the estimate of inte-
grals.Webelievethatthismaterial,sorichinapplicationsandbeauty,isnotgiven
Preface ix
itsdueplaceincoursesinmathematicsorcomputerscience.InChap.12,wehave
madeastrongefforttokeepthingsaseasyaspossible:thechapterisdevotedtothe
first-andsecond-orderEuler–Maclaurinformulas,thusleavingmatterssuchasthe
estimates of the Bernoulli polynomials, the Bernoulli numbers, and the proofs of
every technical detail to the subsequent Chap. 13. It is rather surprising how even
thefirst-orderformula,whoseproofisbasedonasimpleintegrationbyparts,leads
to a primitive version of Stirling’s formula for the approximation of the factorial.
Chapter 14 deals with the approximation of sums of binomials, like the
Ramanujan Q-functions. The proofs here are based on some uniform estimates on
families of sequences and may be skipped by the inexperienced reader.
Finally,alistofthemainformulasandadetailedindexcanbefoundattheend
of the book.
Just to give a taste of the book, here are some of the applications that we
consider and discuss:
(cid:129) Thebirthdayproblem(Example2.30):whatistheprobabilitythattwo(ormore)
people randomly chosen from a group of 25 people have the same birthday?
(cid:129) How to count card shuffles and to perform astonishing magic tricks by means
of the Gilbreath Principle (Sect. 2.3).
(cid:129) The hats problem (Example 4.25): each of the n diners entering a restaurant
leaves his hat at the checkroom. In how many ways can the n hats be redis-
tributed insuchawaythatnoonereceives hisownhat?Ifthetipsyhatchecker
randomlydistributesthehatstotheexitingdiners,whatistheprobabilitythatno
one receives his own hat? Actually, we will be able to solve the more difficult
variantofthe problem whichasks for theprobability thatnoone receives ahat
whose brand is the same as the original one.
(cid:129) TheLeibnizruleforthederivativeofaproductoffunctions(Theorem3.26)and
Faà di Bruno’s formula for the derivatives of a composition of functions
(Theorem5.23)asanapplicationof,respectively,theconceptofsharingwitha
given occupancy and the concept of partition.
(cid:129) The Smith College diploma problem (Example 5.68): at Smith College, diplo-
mas are delivered as follows. The diplomas are randomly distributed to the
graduatingstudents. Thosewho do notreceivetheir own diploma form a circle
and pass the diploma received to their counterclockwise neighbor. Those then
receiving their own diploma leave the circle, while the others form a smaller
circle and repeat the procedure. Determine the probability that exactly k(cid:2)0
hand-offs are necessary before each graduate has his/her own diploma.
(cid:129) The Latin teacher’s random choice (Example 7.30): a teacher wants to select a
studenttotestintheclass.Herandomlyopensabook,sumsupthedigitsofthe
number of the page, and chooses the corresponding student from the alpha-
betical list. What is the probability of being chosen for each student?
(cid:129) The Titus Flavius Josephus problem (Example 9.14) is connected with an
autobiographical episode recounted by the historian Titus Flavius Josephus,
which we now translate into mathematical terms. The problem presents
x Preface
npersonsarrangedinacircle.Havingchosenaninitialpersonandthedirection
ofrotation,onemovesmultipletimesalongthecircle,eliminatingeverysecond
person one meets in the chosen direction until only a single person remains.
Given n(cid:2)1, one seeks todetermine the position of theremainingperson if the
circle is initially formed by n persons.
(cid:129) Givenanywordinanylanguage,howmanysequencesoflettersofaprescribed
lengthdonotcontainthatword?WegivearecursiveanswerinCorollary11.39.
(cid:129) The theorem of the monkey (Theorem 11.43): What is the average number of
chance strokes on a keyboard with m keys necessary to make a given word
appear?
(cid:129) The Conway equation (Theorem 11.53): Two players select distinct sequences
ofequallength‘,withelementsinafixedsetC.Theytossadicewithasmany
faces as the cardinality of C, all labeled with the elements of C, until the last ‘
resultsmatchoneplayer’spattern.Whataretheoddsinfavorofoneortheother
pattern?
(cid:129) Evaluate the mysterious Euler constant c¼ lim H (cid:3)logn up to 12 decimal
n
n!þ1
P1
digits (Example 13.47) and Apéry’s constant fð3Þ¼ 1=k3 up to 16 decimal
k¼1
digits (Example 13.51).
Pn
(cid:129) Faulhaber’s formula (Example 13.22) for the sum km in terms of the first
k¼1
m Bernoulli numbers.
(cid:129) We pick acardfrom adeckof52 playing cards, and we reinsert it inthe deck.
After shuffling, we pick another card and we reinsert it again in the deck. How
many times, onaverage, must we repeat thisprocedure inorder for arepetition
to appear?
The book includes an unusually large collection of examples and problems
whose solutions, as well as corrections and updates, can be found at:
https://discretecalculus.wordpress.com
On many occasions, we have tried to give different proofs of the same result.
This is the case, for instance, for the Inclusion/Exclusion Principle, Faulhaber’s
formula, many combinatorial results, and the estimates for the harmonic numbers
and the binomials. Thus, several paths are possible, depending on the interests
of the reader. Some independent blocks are as follows: the first five chapters on
combinatorics,thesumsandthefinitecalculustechniques(Chap.6),theessentials
of formal power series (Chap. 7) and their applications to the symbolic calculus
(Chap. 11), the basic facts about recurrence relations with an insight into discrete
dynamical systems including Sarkovskii’s theorem (Chap. 9), and the classical
theory on linear recurrences (Chap. 10): here, the alternative resolution method
basedongeneratingformalseriesneedsthecontentofChap.8.Almostallthemain
tools on the approximation of sums appear in Chap. 12, whereas Chap. 13 is
reserved for those who want to master Euler–Maclaurin formulas in the whole
generality,orneedmorethanasecond-orderexpansion.Finally,formostreaders,it