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E-raamat: Handbook of Enumerative Combinatorics

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Presenting the state of the art, the Handbook of Enumerative Combinatorics brings together the work of todays most prominent researchers. The contributors survey the methods of combinatorial enumeration along with the most frequent applications of these methods.

This important new work is edited by Miklós Bóna of the University of Florida where he is a member of the Academy of Distinguished Teaching Scholars. He received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Miklós is the author of four books and more than 65 research articles, including the award-winning Combinatorics of Permutations. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics and Series Editor of the Discrete Mathematics and Its Applications Series for CRC Press/Chapman and Hall.

The first two chapters provide a comprehensive overview of the most frequently used methods in combinatorial enumeration, including algebraic, geometric, and analytic methods. These chapters survey generating functions, methods from linear algebra, partially ordered sets, polytopes, hyperplane arrangements, and matroids. Subsequent chapters illustrate applications of these methods for counting a wide array of objects.

The contributors for this book represent an international spectrum of researchers with strong histories of results. The chapters are organized so readers advance from the more general ones, namely enumeration methods, towards the more specialized ones.

Topics include coverage of asymptotic normality in enumeration, planar maps, graph enumeration, Young tableaux, unimodality, log-concavity, real zeros, asymptotic normality, trees, generalized Catalan paths, computerized enumeration schemes, enumeration of various graph classes, words, tilings, pattern avoidance, computer algebra, and parking functions.

This book will be beneficial to a wide audience. It will appeal to experts on the topic interested in learning more about the finer points, readers interested in a systematic and organized treatment of the topic, and novices who are new to the field.

Arvustused

"Mathematical handbooks are among the most essential library resources, providing compilations of formulas, tables, graphs, etc. Traditional handbooks speak equally to experts and casual users of mathematics. Other handbooks, such as the current work, are really encyclopedic compendiums of survey articles primarily addressing readers who make mathematics their main business. They supplement systematic monographs that develop subjects methodically but require extreme reader commitment and journal literature that provides quick access to specific results for those with prerequisite knowledge. Researchers will benefit from rapid authoritative citations to newer or lesser-known results. Students, undergraduate and graduate, will find accessible, systematic snapshots of whole subjects, helping them discover what they most wish to learn and, equally, what they will then need to learn on the way. Enumerative combinatorics means counting problems, so that subject begins classically with permutations and combinations but is active now with connections to probability, graph theory, statistical mechanics, geometry, representation theory, analysis, and computer science. Chapters here divide between general counting methods, both exact and approximate, and special classes of objects for counting via any suitable means. The volume, part of the 'Discrete Mathematics and Its Applications' series, is well edited by Bóna (Univ. of Florida), who successfully pools the expertise of leaders in the field. Summing up: Recommended. Upper-division undergraduates through professionals/practitioners." D. V. Feldman, University of New Hampshire, Durham, USA, for CHOICE, March 2016

"I cannot think of any topic that I would like to have seen presented here that the book omits. The chapters discuss not only methods in the study of enumerative combinatorics, but also objects that lend themselves to study along these lines. accessible to a wide audience this will clearly be a book that anybody with a serious interest in combinatorics will want to have on his or her bookshelf, and of course it belongs in any self-respecting university library. Having seen firsthand what it takes to edit a handbook like this, I know that Miklós Bóna must have invested a great deal of time and effort in the creation of this volume, as did the authors of the individual chapters. Their efforts have not been in vain; this is a valuable book." MAA Reviews, July 2015

Foreword xix
Preface xxi
Acknowledgments xxiii
I Methods
1(252)
1 Algebraic and Geometric Methods in Enumerative Combinatorics
3(170)
Federico Ardila
1.1 Introduction
5(1)
Part 1 Algebraic Methods
6(1)
1.2 What is a good answer?
6(3)
1.3 Generating functions
9(34)
1.3.1 The ring of formal power series
10(2)
1.3.2 Ordinary generating functions
12(1)
1.3.2.1 Operations on combinatorial structures and their generating functions
13(3)
1.3.2.2 Examples
16(12)
1.3.3 Exponential generating functions
28(1)
1.3.3.1 Operations on labeled structures and their exponential generating functions
28(2)
1.3.3.2 Examples
30(6)
1.3.4 Nice families of generating functions
36(1)
1.3.4.1 Rational generating functions
36(2)
1.3.4.2 Algebraic and D-finite generating functions
38(5)
1.4 Linear algebra methods
43(28)
1.4.1 Determinants in combinatorics
43(1)
1.4.1.1 Preliminaries: Graph matrices
43(1)
1.4.1.2 Walks: the transfer matrix method
44(6)
14.1.3 Spanning trees: the matrix-tree theorem
50(3)
1.4.1.4 Eulerian cycles: the BEST theorem
53(3)
1.4.1.5 Perfect matchings: the Pfaffian method
56(2)
1.4.1.6 Routings: the Lindstrom--Gessel--Viennot lemma
58(7)
1.4.2 Computing determinants
65(1)
1.4.2.1 Is it known?
65(1)
1.4.2.2 Row and column operations
66(1)
1.4.2.3 Identifying linear factors
66(1)
1.4.2.4 Computing the eigenvalues
67(1)
1.4.2.5 LU factorizations
68(1)
1.4.2.6 Hankel determinants and continued fractions
68(3)
1.5 Posets
71(22)
1.5.1 Basic definitions and examples
72(2)
1.5.2 Lattices
74(3)
1.5.3 Zeta polynomials and order polynomials
77(2)
1.5.4 The inclusion-exclusion formula
79(2)
1.5.5 Mobius functions and Mobius inversion
81(1)
1.5.5.1 The Mobius function
81(2)
1.5.5.2 Mobius inversion
83(2)
1.5.5.3 The incidence algebra
85(1)
1.5.5.4 Computing Mobius functions
86(5)
1.5.6 Eulerian posets, flag ƒ-vectors, and flag h-vectors
91(2)
Part 2 Discrete Geometric Methods
93(1)
1.6 Polytopes
93(15)
1.6.1 Basic definitions and constructions
93(5)
1.6.2 Examples
98(2)
1.6.3 Counting faces
100(2)
1.6.4 Counting lattice points: Ehrhart theory
102(6)
1.7 Hyperplane arrangements
108(18)
1.7.1 Basic definitions
109(1)
1.7.2 The characteristic polynomial
110(2)
1.7.3 Properties of the characteristic polynomial
112(1)
1.7.3.1 Deletion and contraction
112(2)
1.7.3.2 Sign alternation and unimodality
114(1)
1.7.3.3 Graphs and proper colorings
114(1)
1.7.3.4 Free arrangements
115(1)
1.7.3.5 Supersolvability
116(1)
1.7.4 Computing the characteristic polynomial
117(8)
1.7.5 The cd-index of an arrangement
125(1)
1.8 Matroids
126(47)
1.8.1 Main example: Vector configurations and linear matroids
127(1)
1.8.2 Basic definitions
128(2)
1.8.3 Examples
130(3)
1.8.4 Basic constructions
133(1)
1.8.5 A few structural results
134(4)
1.8.6 The Tutte polynomial
138(1)
1.8.6.1 Explicit definition
138(1)
1.8.6.2 Recursive definition and universality property
138(2)
1.8.6.3 Activity interpretation
140(1)
1.8.6.4 Finite field interpretation
140(1)
1.8.7 Tutte polynomial evaluations
141(1)
1.8.7.1 General evaluations
142(1)
1.8.7.2 Graphs
143(3)
1.8.7.3 Hyperplane arrangements
146(1)
1.8.7.4 Algebras from arrangements
146(1)
1.8.7.5 Error-correcting codes
147(1)
1.8.7.6 Probability and statistical mechanics
148(1)
1.8.7.7 Other applications
149(1)
1.8.8 Computing the Tutte polynomial
150(3)
1.8.9 Generalizations of the Tutte polynomial
153(2)
1.8.10 Matroid subdivisions, valuations, and the Derksen-Fink invariant
155(18)
2 Analytic Methods
173(80)
Helmut Prodinger
2.1 Introduction
174(1)
2.2 Combinatorial constructions and associated ordinary generating functions
175(5)
2.3 Combinatorial constructions and associated exponential generating functions
180(7)
2.4 Partitions and q-series
187(4)
2.5 Some applications of the adding a slice technique
191(3)
2.6 Lagrange inversion formula
194(2)
2.7 Lattice path enumeration: the continued fraction theorem
196(5)
2.8 Lattice path enumeration: the kernel method
201(4)
2.9 Gamma and zeta function
205(3)
2.10 Harmonic numbers and their generating functions
208(1)
2.11 Approximation of binomial coefficients
209(2)
2.12 Mellin transform and asymptotics of harmonic sums
211(7)
2.13 The Mellin-Perron formula
218(4)
2.14 Mellin-Perron formula: divide-and-conquer recursions
222(1)
2.15 Rice's method
223(4)
2.16 Approximate counting
227(4)
2.17 Singularity analysis of generating functions
231(5)
2.18 Longest runs in words
236(2)
2.19 Inversions in permutations and pumping moments
238(3)
2.20 The tree function
241(3)
2.21 The saddle point method
244(3)
2.22 Hwang's quasi-power theorem
247(6)
II Topics
253(794)
3 Asymptotic Normality in Enumeration
255(26)
E. Rodney Canfield
3.1 Introduction
255(1)
3.2 The normal distribution
256(2)
3.3 Method 1: direct approach
258(4)
3.4 Method 2: negative roots
262(4)
3.5 Method 3: moments
266(4)
3.6 Method 4: singularity analysis
270(2)
3.7 Local limit theorems
272(2)
3.8 Multivariate asymptotic normality
274(2)
3.9 Normality in service to approximate enumeration
276(5)
4 Trees
281(54)
Michael Drmota
4.1 Introduction
281(2)
4.2 Basic notions
283(3)
4.3 Generating functions
286(9)
4.3.1 Generating functions and combinatorial constructions
286(6)
4.3.2 The Lagrange inversion formula
292(1)
4.3.3 The dissymmetry theorem
293(1)
4.3.4 Asymptotics
294(1)
4.4 Unlabeled trees
295(9)
4.4.1 Binary trees
295(4)
4.4.2 Planted plane trees
299(5)
44.3 Unlabeled plane trees
304(6)
4.4.4 General unlabeled trees
305(3)
4.4.5 Simply generated trees and Galton-Watson trees
308(2)
4.5 Labeled trees
310(11)
4.5.1 Cayley trees and related trees
311(5)
4.5.2 Recursive trees
316(2)
4.5.3 Well-labeled trees
318(3)
4.6 Selected topics on trees
321(14)
4.6.1 Spanning trees
321(3)
4.6.2 k-Trees
324(5)
4.6.3 Search trees
329(6)
5 Planar Maps
335(62)
Gilles Schaeffer
5.1 Introduction
336(1)
5.2 What is a map?
336(8)
5.2.1 A few definitions
336(3)
5.2.2 Plane maps, rooted maps and orientations
339(3)
5.2.3 Which maps shall we count?
342(2)
5.3 Counting tree-rooted maps
344(9)
5.3.1 Mullin's decomposition
344(4)
5.3.2 Spanning trees and orientations
348(2)
5.3.3 Vertex blowing and polyhedral nets
350(2)
5.3.4 A summary and some observations
352(1)
5.4 Counting planar maps
353(17)
5.4.1 The exact number of rooted planar maps
353(5)
5.4.2 Unrooted planar maps
358(2)
5.4.3 Two bijections between maps and trees
360(4)
5.4.4 Substitution relations
364(3)
5.4.5 Asymptotic enumeration and uniform random planar maps
367(3)
54.6 Distances in planar maps
370(4)
5.4.7 Local limit, continuum limit
373(1)
5.5 Beyond planar maps, an even shorter account
374(23)
5.5.1 Patterns and universality
375(1)
5.5.2 The bijective canvas and master bijections
376(4)
5.5.3 Maps on surfaces
380(2)
5.5.4 Decorated maps
382(15)
6 Graphs
397(40)
Marc Noy
6.1 Introduction
398(2)
6.2 Graph decompositions
400(6)
6.2.1 Graphs with given connectivity
401(3)
6.2.2 Graphs with given minimum degree
404(1)
6.2.3 Bipartite graphs
405(1)
6.3 Connected graphs with given excess
406(3)
6.4 Regular graphs
409(3)
6.4.1 The pairing model
409(2)
6.4.2 Differential equations
411(1)
6.5 Monotone and hereditary classes
412(5)
6.5.1 Monotone classes
412(2)
6.5.2 Hereditary classes
414(3)
6.6 Planar graphs
417(3)
6.7 Graphs on surfaces and graph minors
420(6)
6.7.1 Graphs on surfaces
420(2)
6.7.2 Graph minors
422(1)
6.7.3 Particular classes
423(3)
6.8 Digraphs
426(2)
6.8.1 Acyclic digraphs
426(2)
6.8.2 Strongly connected digraphs
428(1)
6.9 Unlabeled graphs
428(9)
6.9.1 Counting graphs under symmetries
429(1)
6.9.2 Asymptotics
430(7)
7 Unimodality, Log-concavity, Real-rootedness and Beyond
437(48)
Petter Branden
7.1 Introduction
438(3)
7.2 Probabilistic consequences of real-rootedness
441(1)
7.3 Unimodality and γ-nonnegativity
442(8)
7.3.1 An action on permutations
443(3)
7.3.2 γ-nonnegativity of h-polynomials
446(1)
7.3.3 Barycentric subdivisions
447(2)
7.3.4 Unimodality of h*-polynomials
449(1)
7.4 Log-concavity and matroids
450(2)
7.5 Infinite log-concavity
452(1)
7.6 The Neggers--Stanley conjecture
453(3)
7.7 Preserving real-rootedness
456(4)
7.7.1 The subdivision operator
459(1)
7.8 Common interleavers
460(8)
7.8.1 s-Eulerian polynomials
464(1)
7.8.2 Eulerian polynomials for finite Coxeter groups
465(3)
7.9 Multivariate techniques
468(8)
7.9.1 Stable polynomials and matroids
469(1)
7.9.2 Strong Rayleigh measures
470(2)
7.9.3 The symmetric exclusion process
472(2)
7.9.4 The Grace--Walsh--Szego theorem, and the proof of Theorem 7.5.1
474(2)
7.10 Historical notes
476(9)
8 Words
485(56)
Dominique Perrin
Antonio Restivo
8.1 Introduction
486(1)
8.2 Preliminaries
487(5)
8.2.1 Generating series
488(3)
8.2.2 Automata
491(1)
8.3 Conjugacy
492(8)
8.3.1 Periods
492(1)
8.3.2 Necklaces
493(4)
8.3.3 Circular codes
497(3)
8.4 Lyndon words
500(4)
8.4.1 The factorization theorem
501(2)
8.4.2 Generating Lyndon words
503(1)
8.5 Eulerian graphs and de Bruijn cycles
504(8)
8.5.1 The BEST theorem
507(1)
8.5.2 The Matrix-tree theorem
508(2)
8.5.3 Lyndon words and de Bruijn cycles
510(2)
8.6 Unavoidable sets
512(9)
8.6.1 Algorithms
513(3)
8.6.2 Unavoidable sets of constant length
516(3)
8.6.3 Conclusion
519(2)
8.7 The Burrows--Wheeler transform
521(4)
8.7.1 The inverse transform
522(2)
8.7.2 Descents of a permutation
524(1)
8.8 The Gessel-Reutenauer bijection
525(5)
8.8.1 Gessel-Reutenauer bijection and de Bruijn cycles
527(3)
8.9 Suffix arrays
530(11)
8.9.1 Suffix arrays and Burrows-Wheeler transform
530(2)
8.9.2 Counting suffix arrays
532(9)
9 Tilings
541(48)
James Propp
9.1 Introduction
541(7)
9.2 The transfer matrix method
548(3)
9.3 Other determinant methods
551(5)
9.3.1 The path method
551(2)
9.3.2 The permanent-determinant and Hafnian-Pfaffian method
553(2)
9.3.3 The spanning tree method
555(1)
9.4 Representation-theoretic methods
556(3)
9.5 Other combinatorial methods
559(3)
9.6 Related topics, and an attempt at history
562(3)
9.7 Some emergent themes
565(6)
9.7.1 Recurrence relations
565(1)
9.7.2 Smoothness
566(2)
9.7.3 Non-periodic weights
568(1)
9.7.4 Other numerical patterns
569(1)
9.7.5 Symmetry
570(1)
9.8 Software
571(2)
9.9 Frontiers
573(16)
10 Lattice Path Enumeration
589(90)
Christian Krattenthaler
10.1 Introduction
589(3)
10.2 Lattice paths without restrictions
592(2)
10.3 Linear boundaries of slope 1
594(4)
10.4 Simple paths with linear boundaries of rational slope, I
598(8)
10.5 Simple paths with linear boundaries with rational slope, II
606(5)
10.6 Simple paths with a piecewise linear boundary
611(1)
10.7 Simple paths with general boundaries
612(3)
10.8 Elementary results on Motzkin and Schroder paths
615(3)
10.9 A continued fraction for the weighted counting of Motzkin paths
618(4)
10.10 Lattice paths and orthogonal polynomials
622(7)
10.11 Motzkin paths in a strip
629(4)
10.12 Further results for lattice paths in the plane
633(5)
10.13 Non-intersecting lattice paths
638(13)
10.14 Lattice paths and their turns
651(6)
10.15 Multidimensional lattice paths
657(1)
10.16 Multidimensional lattice paths bounded by a hyperplane
658(1)
10.17 Multidimensional paths with a general boundary
658(1)
10.18 The reflection principle in full generality
659(8)
10.19 q-Counting of lattice paths and Rogers-Ramanujan identities
667(3)
10.20 Self-avoiding walks
670(9)
11 Catalan Paths and g, t-enumeration
679(74)
James Haglund
11.1 Introduction
680(1)
11.2 Introduction to q-analogues and Catalan numbers
680(22)
11.2.1 Permutation statistics and Gaussian polynomials
680(6)
11.2.2 The Catalan numbers and Dyck paths
686(3)
11.2.3 The q-Vandermonde convolution
689(2)
11.2.4 Symmetric functions
691(1)
11.2.4.1 The basics
691(3)
11.2.4.2 Tableaux and Schur functions
694(3)
11.2.4.3 Statistics on tableaux
697(1)
11.2.5 Representation theory
698(3)
11.2.5.1 The ring of coinvariants and the space of diagonal harmonics
701(1)
11.3 The q, t-Catalan numbers
702(14)
11.3.1 The bounce statistic
704(3)
11.3.2 The special values t = 1 and t = 1/q
707(2)
11.3.3 The symmetry problem and the dinv statistic
709(3)
11.3.4 q-Lagrange inversion
712(4)
11.4 Parking functions and the Hilbert series
716(15)
11.4.1 Extension of the dinv statistic
716(2)
11.4.2 An explicit formula
718(3)
11.4.3 The statistic area'
721(1)
11.4.4 The pmaj statistic
722(3)
11.4.5 The cyclic-shift operation
725(4)
11.4.6 Tesler matrices
729(2)
11.5 The q, t-Schroder polynomial
731(11)
11.5.1 The Schroder bounce and area statistics
731(3)
11.5.2 Recurrences and explicit formulae
734(4)
11.5.3 The special value t = 1/q
738(1)
11.5.4 A Schroder dinv-statistic
739(1)
11.5.5 The shuffle conjecture
740(2)
11.6 Rational Catalan combinatorics
742(11)
11.6.1 The superpolynomial invariant of a torus knot
742(3)
11.6.2 Tesler matrices and the superpolynomial
745(8)
12 Permutation Classes
753(82)
Vincent Vatter
12.1 Introduction
754(17)
12.1.1 Basics
755(4)
12.1.2 Avoiding a permutation of length three
759(2)
12.1.3 Wilf-equivalence
761(5)
12.1.4 Avoiding a longer permutation
766(5)
12.2 Growth rates of principal classes
771(15)
12.2.1 Matrices and the interval minor order
773(4)
12.2.2 The number of light matrices
777(1)
12.2.3 Matrices avoiding Jk are light
778(2)
12.2.4 Dense matrices contain many permutations
780(2)
12.2.5 Dense matrices avoiding Jk
782(4)
12.3 Notions of structure
786(14)
12.3.1 Merging and splitting
788(4)
12.3.2 The substitution decomposition
792(5)
12.3.3 Atomicity
797(3)
12.4 The set of all growth rates
800(18)
12.4.1 Monotone grid classes
803(5)
12.4.2 Geometric grid classes
808(7)
12.4.3 Generalized grid classes
815(3)
124.4 Small permutation classes
818(17)
13 Parking Functions
835(60)
Catherine H. Yan
13.1 Introduction
836(1)
13.2 Parking functions and labeled trees
837(13)
13.2.1 Labeled trees with Prufer code
837(2)
13.2.2 Inversions of labeled trees
839(4)
13.2.3 Graph searching algorithms
843(4)
13.2.4 External activity of labeled trees
847(1)
13.2.5 Sparse connected graphs
848(2)
13.3 Many faces of parking functions
850(9)
13.3.1 Hashing and linear probing
850(2)
13.3.2 Lattice of noncrossing partitions
852(3)
13.3.3 Hyperplane arrangements
855(2)
13.3.4 Allowable input-output pairs in a priority queue
857(1)
13.3.5 Two variations of parking functions
858(1)
13.4 Generalized parking functions
859(15)
13.4.1 u-parking functions
859(2)
13.4.2 A parking polytope
861(3)
13.4.3 Theory of Goncarov polynomials
864(6)
13.4.4 Classical parking functions
870(4)
13.5 Parking functions associated with graphs
874(10)
13.5.1 G-parking functions
874(1)
13.5.2 Abelian sandpile model
875(3)
13.5.3 Multiparking functions, graph searching, and the Tutte polynomial
878(6)
13.6 Final remarks
884(11)
14 Standard Young Tableaux
895(80)
Ron Adin
Yuval Roichman
14.1 Introduction
896(2)
14.1.1 Appetizer
896(2)
14.1.2 General
898(1)
14.2 Preliminaries
898(7)
14.2.1 Diagrams and tableaux
898(1)
14.2.2 Connectedness and convexity
899(1)
14.2.3 Invariance under symmetry
900(1)
14.2.4 Ordinary, skew and shifted shapes
901(2)
14.2.5 Interpretations
903(1)
14.2.5.1 The Young lattice
903(1)
14.2.5.2 Ballot sequences and lattice paths
903(1)
14.2.5.3 The order polytope
904(1)
14.2.5.4 Other interpretations
905(1)
14.2.6 Miscellanea
905(1)
14.3 Formulas for thin shapes
905(3)
14.3.1 Hook shapes
905(1)
14.3.2 Two-rowed shapes
906(1)
14.3.3 Zigzag shapes
907(1)
14.4 Jeu de taquin and the RS correspondence
908(5)
14.4.1 Jeu de taquin
908(2)
14.4.2 The Robinson-Schensted correspondence
910(1)
14.4.3 Enumerative applications
911(2)
14.5 Formulas for classical shapes
913(6)
14.5.1 Ordinary shapes
913(3)
14.5.2 Skew shapes
916(2)
14.5.3 Shifted shapes
918(1)
14.6 More proofs of the hook length formula
919(11)
14.6.1 A probabilistic proof
919(2)
14.6.2 Bijective proofs
921(5)
14.6.3 Partial difference operators
926(4)
14.7 Formulas for skew strips
930(8)
14.7.1 Zigzag shapes
930(3)
14.7.2 Skew strips of constant width
933(5)
14.8 Truncated and other non-classical shapes
938(6)
14.8.1 Truncated shifted staircase shape
938(1)
14.8.2 Truncated rectangular shapes
939(2)
14.8.3 Other truncated shapes
941(1)
14.8.4 Proof approaches for truncated shapes
942(2)
14.9 Rim hook and domino tableaux
944(5)
14.9.1 Definitions
944(1)
14.9.2 The r-quotient and r-core
945(4)
14.10 q-Enumeration
949(8)
14.10.1 Permutation statistics
949(1)
14.10.2 Statistics on tableaux
950(2)
14.10.3 Thin shapes
952(1)
14.10.3.1 Hook shapes
952(1)
14.10.3.2 Zigzag shapes
952(1)
14.10.3.3 Two-rowed shapes
953(2)
14.10.4 The general case
955(1)
14.10.4.1 Counting by descents
955(1)
14.10.4.2 Counting by major index
956(1)
14.10.4.3 Counting by inversions
957(1)
14.11 Counting reduced words
957(4)
14.11.1 Coxeter generators and reduced words
957(1)
14.11.2 Ordinary and skew shapes
958(2)
14.11.3 Shifted shapes
960(1)
14.12 Appendix 1: Representation theoretic aspects
961(3)
14.12.1 Degrees and enumeration
961(2)
14.12.2 Characters and q-enumeration
963(1)
14.13 Appendix 2: Asymptotics and probabilistic aspects
964(11)
15 Computer Algebra
975(72)
Manuel Kauers
15.1 Introduction
976(1)
15.2 Computer algebra essentials
977(19)
15.2.1 Numbers
978(1)
15.2.1.1 Integers and rational numbers
978(1)
15.2.1.2 Algebraic numbers
978(1)
15.2.1.3 Real and complex numbers
979(1)
15.2.1.4 Finite fields and modular arithmetic
980(1)
15.2.1.5 Lattice reduction
981(2)
15.2.2 Polynomials
983(1)
15.2.2.1 Arithmetic and factorization
984(1)
15.2.2.2 Evaluation and interpolation
985(1)
15.2.2.3 Grobner bases
985(2)
15.2.2.4 Cylindrical algebraic decomposition
987(1)
15.2.3 Formal power series
988(1)
15.2.3.1 Truncated power series
988(1)
15.2.3.2 Lazy power series
989(1)
15.2.3.3 Generalized series
990(1)
15.2.3.4 The multivariate case
990(1)
15.2.4 Operators
991(1)
15.2.4.1 Ore algebras
992(1)
15.2.4.2 Actions and solutions
993(3)
15.3 Counting algorithms
996(15)
15.3.1 Special purpose algorithms
996(3)
15.3.2 Combinatorial species
999(1)
15.3.2.1 Formal definition and associated series
1000(1)
15.3.2.2 Standard constructions
1000(2)
15.3.2.3 Recursive specifications
1002(1)
15.3.3 Partition analysis
1003(1)
15.3.3.1 Ω-Calculus
1003(2)
15.3.3.2 Ehrhart theory
1005(2)
15.3.4 Computational group theory
1007(1)
15.3.4.1 Permutation groups
1007(1)
15.3.4.2 Finitely presented groups
1008(2)
15.3.5 Software
1010(1)
15.4 Symbolic summation
1011(17)
15.4.1 Classical algorithms
1011(1)
15.4.1.1 Hypergeometric terms
1011(1)
15.4.1.2 Gosper's algorithm
1012(1)
15.4.1.3 Zeilberger's algorithm
1013(1)
15.4.1.4 Petkovsek's algorithm
1014(1)
15.4.2 ΠΣ-Theory
1015(1)
15.4.2.1 ΠΣ-Expressions and difference fields
1015(2)
15.4.2.2 Indefinite summation
1017(2)
15.4.2.3 Creative telescoping
1019(1)
15.4.2.4 D'Alembertian solutions
1020(1)
15.4.2.5 Nested definite sums
1021(1)
15.4.3 The holonomic systems approach
1021(1)
15.4.3.1 D-finite and holonomic functions
1022(1)
15.4.3.2 Closure properties
1023(2)
15.4.3.3 Summation and integration
1025(1)
15.4.3.4 Nested sums and integrals
1026(1)
15.4.4 Implementations and current research topics
1027(1)
15.5 The guess-and-prove paradigm
1028(19)
Index 1047
Miklós Bóna received his Ph.D. in mathematics at Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where in 2010 he was inducted in the Academy of Distinguished Teaching Scholars. 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 the Outstanding Title Award from Choice, the journal of the American Library Association. He has mentored numerous graduate and undergraduate students. Miklós Bóna is an editor-in-chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.
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