Introduction to Enumerative and Analytic Combinatorics

These award-winning textbook targets the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The author's goal is to make combinatorics more accessible to encourage student interest and to expand the number of students studying this rapidly expanding field.

Autorentext

Miklós Bóna received his Ph.D in mathematics from the Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where, in 2010, he was inducted into the Academy of Distinguished Teaching Scholars. Professor Bóna has mentored numerous graduate and undergraduate students. He is the author of four books and more than 65 research articles, mostly focusing on enumerative and analytic combinatorics. His book, Combinatorics of Permutations, won a 2006 Outstanding Title Award from Choice, the journal of the American Library Association. He is also an Editor-in-Chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.

Inhalt

Basic methodsWhen we add and when we subtractWhen we multiplyWhen we divideApplications of basic counting principlesThe pigeonhole principleNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

Applications of basic methodsMultisets and compositionsSet partitionsPartitions of integersThe inclusion-exclusion principleThe twelvefold wayNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

Generating functionsPower seriesWarming up: Solving recurrence relationsProducts of generating functionsCompositions of generating functionsA different type of generating functionsNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

TOPICS

Counting permutationsEulerian numbersThe cycle structure of permutationsCycle structure and exponential generating functionsInversionsAdvanced applications of generating functions to permutation enumerationNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

Counting graphsTrees and forestsGraphs and functionsWhen the vertices are not freely labeledGraphs on colored verticesGraphs and generating functionsNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

Extremal combinatoricsExtremal graph theoryHypergraphsSomething is more than nothing: Existence proofsNotesChapter reviewExercisesSolutions to exercises****Supplementary exercises

AN ADVANCED METHOD

Analytic combinatoricsExponential growth ratesPolynomial precision
More precise asymptoticsNotesChapter reviewExercisesSolutions to exercisesSupplementary exercises

SPECIAL TOPICS

Symmetric structuresDesignsFinite projective planesError-correcting codesCounting symmetric structuresNotesChapter reviewExercisesSolutions to exercisesSupplementary exercises

Sequences in combinatoricsUnimodality
Log-concavity
The real zeros property
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises

Counting magic squares and magic cubesA distribution problem
Magic squares of fixed size
Magic squares of fixed line sum
Why magic cubes are different
Notes
Chapter review
Exercises
Solutions to exercises
Supplementary exercises

Appendix: The method of mathematical induction
Weak induction
Strong induction