Fibonacci and Catalan numbers

382 Pages·2012·4.685 MB·English

by  Grimaldi, Ralph P

FIBONACCI AND CATALAN NUMBERS

FIBONACCI AND
CATALAN NUMBERS
AN INTRODUCTION
Ralph P. Grimaldi
Rose-Hulman Institute of Technology
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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:TheBinetFormforF 51
n
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