Fibonacci and catalan numbers : an introduction

374 Pages·2012·1.59 MB·English

by  Ralph P Grimaldi, Wiley InterScience (Online service)

FIBONACCI AND CATALAN NUMBERS
FIBONACCI AND
CATALAN NUMBERS
AN INTRODUCTION
Ralph P. Grimaldi
Rose-Hulman Institute of Technology
Copyright©2012byJohnWiley&Sons,Inc.Allrightsreserved
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Grimaldi,RalphP.
Fibonacciandcatalannumbers:anintroduction/RalphP.Grimaldi.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-63157-7
1. Fibonaccinumbers. 2. Recurrentsequences(Mathematics) 3. Catalannumbers
(Mathematics) 4. Combinatorialanalysis. I. Title.
QA241.G7252012
512.7’2–dc23
2011043338
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Dedicated to the Memory of
Josephine and Joseph
and
Mildred and John
and
Madge
CONTENTS
PREFACE xi
PARTONE THEFIBONACCINUMBERS
1. HistoricalBackground 3
2. TheProblemoftheRabbits 5
3. TheRecursiveDefinition 7
4. PropertiesoftheFibonacciNumbers 8
5. SomeIntroductoryExamples 13
6. CompositionsandPalindromes 23
7. Tilings:DivisibilityPropertiesoftheFibonacciNumbers 33
8. ChessPiecesonChessboards 40
9. Optics,Botany,andtheFibonacciNumbers 46
10. SolvingLinearRecurrenceRelations:TheBinetFormforFn 51
11. Moreonαandβ:ApplicationsinTrigonometry,Physics,Continued
Fractions,Probability,theAssociativeLaw,andComputerScience 65
12. ExamplesfromGraphTheory:AnIntroductiontotheLucasNumbers 79
13. TheLucasNumbers:FurtherPropertiesandExamples 100
14. Matrices,TheInverseTangentFunction,andanInfiniteSum 113
15. ThegcdPropertyfortheFibonacciNumbers 121
vii
viii CONTENTS
16. AlternateFibonacciNumbers 126
17. OneFinalExample? 140
PARTTWO THECATALANNUMBERS
18. HistoricalBackground 147
19. AFirstExample:AFormulafortheCatalanNumbers 150
20. SomeFurtherInitialExamples 159
21. DyckPaths,Peaks,andValleys 169
22. YoungTableaux,Compositions,andVerticesandArcs 183
23. TriangulatingtheInteriorofaConvexPolygon 192
24. SomeExamplesfromGraphTheory 195
25. PartialOrders,TotalOrders,andTopologicalSorting 205
26. SequencesandaGeneratingTree 211
27. MaximalCliques,aComputerScienceExample,andtheTennisBall
Problem 219
28. TheCatalanNumbersatSportingEvents 226
29. ARecurrenceRelationfortheCatalanNumbers 231
30. TriangulatingtheInteriorofaConvexPolygonfortheSecondTime 236
31. RootedOrderedBinaryTrees,PatternAvoidance,andData
Structures 238
32. Staircases,ArrangementsofCoins,TheHandshakingProblem,and
NoncrossingPartitions 250
33. TheNarayanaNumbers 268
34. RelatedNumberSequences:TheMotzkinNumbers,
TheFineNumbers,andTheSchro¨derNumbers 282
CONTENTS ix
35. GeneralizedCatalanNumbers 290
36. OneFinalExample? 296
SolutionsfortheOdd-NumberedExercises 301
Index 355
PREFACE
InJanuaryof1992,Ipresentedaminicourseatthejointnationalmathematicsmeetings
heldthatyearinBaltimore,Maryland.Theminicoursehadbeenapprovedbyacom-
mitteeoftheMathematicalAssociationofAmerica—themissionofthatcommittee
beingtheevaluationofproposedminicourses.Inthiscase,theminicoursewasespe-
ciallypromotedbyProfessorFredHoffmanofFloridaAtlanticUniversity.Presented
intwotwo-hoursessions,thefirstsessionoftheminicoursetoucheduponexamples,
properties,andapplicationsofthesequenceofFibonaccinumbers.Thesecondpart
investigated comparable ideas for the sequence of Catalan numbers. The audience
was comprised primarily of college and university mathematics professors, along
withasubstantialnumberofgraduatestudentsandundergraduatestudents,aswell
assomemathematicsteachersfromhighschoolsintheBaltimoreandWashington,
D.C.areas.
Sinceitsfirstpresentation,thecoverageinthisminicoursehasexpandedoverthe
past19years,asIdeliveredthematerialnineadditionaltimesatlaterjointnational
mathematicsmeetings—thelatestbeingthemeetingsheldinJanuaryof2010inSan
Francisco. In addition, the topics have also been presented completely, or in part,
at more than a dozen state sectional meetings of the Mathematical Association of
America and at several workshops, where, on occasion, some high school students
wereinattendance.Evaluationsprovidedbythosewhoattendedthelecturesdirected
metofurtherrelevantmaterialandalsohelpedtoimprovethepresentations.
At all times, the presentations were developed so that everyone in the audience
wouldbeabletounderstandatleastsome,ifnotasubstantialamount,ofthemate-
rial. Consequently, this resulting book, which has grown out of these experiences,
shouldbelookeduponasanintroduction tothemanyinterestingproperties,exam-
ples,andapplicationsthatariseinthestudyoftwoofthemostfascinatingsequencesof
numbers.Asweprogressthroughthevariouschapters,weshouldsooncometounder-
standwhythesesequencesareoftenreferredtoasubiquitous,especiallyincourses
indiscretemathematicsandcombinatorics,wheretheyappearsoveryoften.Forthe
Fibonaccinumbers,weshallfindapplicationsinsuchdiverseareasassettheory,the
compositions of integers, graph theory, matrix theory, trigonometry, botany, chem-
istry,physics,probability,andcomputationalcomplexity.WeshallfindtheCatalan
numbersariseinsituationsdealingwithlatticepaths,graphtheory,geometry,partial
orders,sequences,patternavoidance,partitions,computerscience,andevensporting
events.
xi
xii PREFACE
FEATURES
Followingarebriefdescriptionsoffourofthemajorfeaturesofthisbook.
1. UsefulResources
Thebookcanbeusedinavarietyofways:
(i) AsatextbookforanintroductorycourseontheFibonaccinumbersand/or
theCatalannumbers.
(ii) Asasupplementforacourseindiscretemathematicsorcombinatorics.
(iii) As a source for students seeking a topic for a research paper or some
othertypeofprojectinamathematicalareatheyhavenotcovered,oronly
brieflycovered,inaformalmathematicscourse.
(iv) Asasourceforindependentstudy.
2. Organization
The book is divided into 36 chapters. The first 17 chapters constitute Part
One of the book and deal with the Fibonacci numbers. Chapters 18 through
36comprisePartTwo,whichcoversthematerialontheCatalannumbers.The
twopartscanbecoveredineitherorder.InPartTwo,somereferencesaremade
tomaterialinPartOne.Theseareusuallyonlycomparisons.Shouldtheneed
arise, one can readily find the material from Part One that is mentioned in
conjunctionwithsomethingcoveredinPartTwo.
Furthermore, each of Parts One and Two ends with a bibliography. These
referencesshouldproveusefulforthereaderinterestedinlearningevenmore
abouteitherofthesetworatheramazingnumbersequences.
3. DetailedExplanations
Sincethisbookistoberegardedasanintroduction,examplesand,especially,
proofsarepresentedwithdetailedexplanations.Suchexamplesandproofsare
designed to be careful and thorough. Throughout the book, the presentation
isfocusedprimarilyonimprovingunderstandingforthereaderwhoisseeing
most,ifnotall,ofthismaterialforthefirsttime.
In addition, every attempt has been made to provide any necessary back-
groundmaterial,wheneverneeded.
4. Exercises
Thereareover300exercisesthroughoutthebook.Theseexercisesarepri-
marily designed to review the basic ideas provided in a given chapter and to
introduceadditionalpropertiesandexamples.Insomecases,theexercisesalso
extend what is covered in one or more of the chapters. Answers for all the
odd-numberedexercisesareprovidedatthebackofthebook.
ANCILLARY
There is an Instructor’sSolutionManual that is available for those instructors who
adoptthisbook.Themanualcanbeobtainedfromthepublisherviawrittenrequest