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Combinatorics of Set Partitions [Kõva köide]

(University of Haifa, Israel)
  • Formaat: Hardback, 516 pages, kõrgus x laius: 234x156 mm, kaal: 1110 g, 46 Tables, black and white; 38 Illustrations, black and white
  • Ilmumisaeg: 27-Jul-2012
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1439863334
  • ISBN-13: 9781439863336
Teised raamatud teemal:
Combinatorics of Set PartitionsSuurem pilt
  • Formaat: Hardback, 516 pages, kõrgus x laius: 234x156 mm, kaal: 1110 g, 46 Tables, black and white; 38 Illustrations, black and white
  • Ilmumisaeg: 27-Jul-2012
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1439863334
  • ISBN-13: 9781439863336
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781439863336.html
Märksõnad:
Focusing on a very active area of mathematical research in the last decade, Combinatorics of Set Partitions presents methods used in the combinatorics of pattern avoidance and pattern enumeration in set partitions. Designed for students and researchers in discrete mathematics, the book is a one-stop reference on the results and research activities of set partitions from 1500 A.D. to today.

Each chapter gives historical perspectives and contrasts different approaches, including generating functions, kernel method, block decomposition method, generating tree, and Wilf equivalences. Methods and definitions are illustrated with worked examples and Maple code. End-of-chapter problems often draw on data from published papers and the authors extensive research in this field. The text also explores research directions that extend the results discussed. C++ programs and output tables are listed in the appendices and available for download on the authors web page.

Arvustused

"Containing 375 references, this book works best as an encyclopedia for researchers working in this topic. But it can also be effectively used by students who are interested in the combinatorics of set partitions and want to get a hand on researching them. a very useful reference for researchers of the enumerative side of set partitions." Acta Sci. Math. (Szeged), 80, 2014

" a comprehensive account of the history and current research in the combinatorics of pattern enumeration and pattern avoidance in set partitions. While it is aimed primarily at advanced undergraduate and graduate students in discrete mathematics with a focus on set partitions, its extensive bibliography, with 375 entries, and the variety of constructions and approaches used in the text make it a valuable reference for researchers in this field." Ricardo Mamede, Zentralblatt MATH 1261

" a comprehensive account of recent and current research on the pattern-related aspects of set partitions." David Callan, Mathematical Reviews, April 2013

Preface xix
Acknowledgment xxv
Author Biographies xxvii
1 Introduction 1(30)
1.1 Historical Overview and Earliest Results
1(9)
1.2 Timeline of Research for Set Partitions
10(10)
1.3 A More Detailed Book
20(7)
1.3.1 Basic Results
20(1)
1.3.2 Statistics on Set Partitions
21(2)
1.3.3 Pattern Avoidance in Set Partitions
23(2)
1.3.4 Restricted Set Partitions
25(1)
1.3.5 Asymptotics Results on Set Partitions
25(1)
1.3.6 Generating Set Partitions
26(1)
1.3.7 Normal Ordering and Set Partitions
27(1)
1.4 Exercises
27(4)
2 Basic Tools of the Book 31(44)
2.1 Sequences
31(8)
2.2 Solving Recurrence Relations
39(6)
2.2.1 Guess and Check
39(1)
2.2.2 Iteration
40(1)
2.2.3 Characteristic Polynomial
40(5)
2.3 Generating Functions
45(18)
2.4 Lagrange Inversion Formula
63(1)
2.5 The Principle of Inclusion and Exclusion
64(3)
2.6 Generating Trees
67(4)
2.7 Exercises
71(4)
3 Preliminary Results on Set Partitions 75(32)
3.1 Dobinski's Formula
75(9)
3.2 Different Representations
84(15)
3.2.1 Block Representation
85(4)
3.2.2 Circular Representation and Line Diagram
89(1)
3.2.3 Flattened Set Partitions
90(5)
3.2.4 More Representations
95(14)
3.2.4.1 Canonical Representation
95(1)
3.2.4.2 Graphical Representation
96(1)
3.2.4.3 Standard Representation
96(2)
3.2.4.4 Rook Placement Representation
98(1)
3.3 Exercises
99(4)
3.4 Research Directions and Open Problems
103(4)
4 Subword Statistics on Set Partitions 107(58)
4.1 Subword Patterns of Size Two: Rises, Levels and Descents
109(10)
4.1.1 Number Levels
114(3)
4.1.2 Nontrivial Rises and Descents
117(2)
4.2 Peaks and Valleys
119(12)
4.2.1 Counting Peaks in Words
120(3)
4.2.2 Counting Peaks
123(3)
4.2.3 Counting Valleys
126(5)
4.3 Subword Patterns: l-Rises, l-Levels, and l-Descents
131(10)
4.3.1 Long- Rise Pattern
131
4.3.2 Long-Level Pattern
130(9)
4.3.3 Long-Descent. Pattern
139(2)
4.4 Families of Subword Patterns
141(11)
4.4.1 The Patterns 122...2, 11...12
141(2)
4.4.2 The Patterns 22...21, 211...1
143(3)
4.4.3 The Pattern mpm
146(2)
4.4.4 The Pattern mp(m+1)
148(3)
4.4.5 The Pattern (m+1)pm
151(1)
4.5 Patterns of Size Three
152(6)
4.6 Exercises
158(1)
4.7 Research Directions and Open Problems
159(6)
5 Nonsubword Statistics on Set Partitions 165(58)
5.1 Statistics and Block Representation
167(2)
5.2 Statistics and Canonical and Rook Representations
169(6)
5.3 Records and Weak Records
175(7)
5.3.1 Weak Records
176(3)
5.3.2 Sum of Positions of Records
179(2)
5.3.3 Sum of Positions of Additional Weak Records
181(1)
5.4 Number of Positions Between Adjacent Occurrences of a Letter
182(14)
5.4.1 The Statistic dis
185(5)
5.4.2 The Statistic m-Distance
190(3)
5.4.3 Combinatorial Proofs
193(3)
5.5 The Internal Statistic
196(6)
5.5.1 The Statistic int
198(2)
5.5.2 The Statistic int1
200(2)
5.6 Statistics and Generalized Patterns
202(7)
5.7 Major Index
209(6)
5.8 Number of Crossings, Nestings and Alignments
215(1)
5.9 Exercises
216(3)
5.10 Research Directions and Open Problems
219(4)
6 Avoidance of Patterns in Set Partitions 223(84)
6.1 History and Connections
223(2)
6.2 Avoidance of Subsequence Patterns
225(50)
6.2.1 Pattern-Avoiding Fillings of Diagrams
226(10)
6.2.2 Basic Facts and Patterns of Size Three
236(1)
6.2.3 Noncrossing and Nonnesting Set Partitions
237(2)
6.2.4 The Patterns 12...(k + 1)12...k, 12...k12...(k + 1)
239(4)
6.2.4.1 The Pattern 12...(k+1)12...k
240(1)
6.2.4.2 The Pattern 12...k12...(k+1)
241(1)
6.2.4.3 The Equivalence
242(1)
6.2.5 Patterns of the Form 1(τ + 1)
243(1)
6.2.6 The Patterns 12...k1, 12...k12
244(6)
6.2.7 Patterns Equivalent to 12k...m
250(1)
6.2.8 Binary Patterns
250(6)
6.2.9 Patterns Equivalent to 12k13
256(3)
6.2.10 Landscape Patterns
259(5)
6.2.11 Patterns of Size Four
264(4)
6.2.11.1 The Pattern 1123
264(3)
6.2.11.2 Classification of Patterns of Size Four
267(1)
6.2.12 Patterns of Size Five
268(2)
6.2.12.1 The Equivalence 12112 ~ 12212
268(2)
6.2.12.2 Classification of Patterns of Size Five
270(1)
6.2.13 Patterns of Size Six
270(2)
6.2.14 Patterns of Size Seven
272(3)
6.3 Generalized Patterns
275(14)
6.3.1 Patterns of Type (1, 2)
276(4)
6.3.2 Patterns of Type (2, 1)
280(9)
6.4 Partially Ordered Patterns
289(9)
6.4.1 Patterns of Size Three
290(4)
6.4.2 Shuffle Patterns
294(4)
6.5 Exercises
298(2)
6.6 Research Directions and Open Problems
300(7)
7 Multi Restrictions on Set Partitions 307(72)
7.1 Avoiding a Pattern of Size Three and Another Pattern
308(7)
7.1.1 The Patterns 112,121
309(3)
7.1.2 The Pattern 123
312(2)
7.1.3 The Pattern 122
314(1)
7.2 Pattern Avoidance in Noncrossing Set Partitions
315(12)
7.2.1 CC-Equivalences
320(2)
7.2.2 Generating Functions
322(4)
7.2.3 CC-Equivalences of Patterns of Size Four
326(1)
7.2.4 CC-Equivalences of Patterns of Size Five
326(1)
7.3 General Equivalences
327(3)
7.4 Two Patterns of Size Four
330(4)
7.5 Left Motzkin Numbers
334(5)
7.5.1 The Pairs (1222, 1212) and (1112, 1212)
335(1)
7.5.2 The Pair (1211,1221)
336(1)
7.5.3 The Pair (1222, 1221)
337(2)
7.6 Sequence A054391
339(7)
7.6.1 The Pairs (1212, 12221), (1212, 11222), (1212, 11122)
340(1)
7.6.2 The Pair (1221, 12311)
340(2)
7.6.3 The Pairs (1221,12112), (1221, 12122)
342(4)
7.7 Catalan and Generalized Catalan Numbers
346(1)
7.8 Pell Numbers
347(7)
7.8.1 Counting Pn(1211, 1212, 1213) by inv
348(3)
7.8.2 Counting Pn(1212, 1222, 1232) by Comaj
351(3)
7.9 Regular Set Partitions
354(5)
7.9.1 Noncrossing Regular Set Partitions
357(1)
7.9.2 Nonnesting Regular Set Partitions
358(1)
7.10 Distance Restrictions
359(3)
7.11 Singletons
362(2)
7.12 Block-Connected
364(3)
7.13 Exercises
367(5)
7.14 Research Directions and Open Problems
372(7)
8 Asymptotics and Random Set Partition 379(44)
8.1 Tools from Probability Theory
380(7)
8.2 Tools from Complex Analysis
387(7)
8.3 Z-statistics
394(7)
8.4 Set Partitions as Geometric Words
401(4)
8.5 Asymptotics for Set Partitions
405(13)
8.5.1 Asymptotics for Bell and Stirling Numbers
405(11)
8.5.1.1 On the Number of Blocks
409(2)
8.5.1.2 On the Number of Distinct Block Sizes
411(5)
8.5.2 On Number of Blocks in a Noncrossing Set Partition
416(1)
8.5.3 Records
417(1)
8.6 Exercises
418(2)
8.7 Research Directions and Open Problems
420(3)
9 Gray Codes, Loopless Algorithms and Set Partitions 423(16)
9.1 Gray Code and Loopless Algorithms
424(4)
9.2 Gray Codes for Pn
428(6)
9.3 Loopless Algorithm for Generating
434(2)
9.4 Exercises
436(1)
9.5 Research Directions and Open Problems
437(2)
10 Set Partitions and Normal Ordering 439(34)
10.1 Preliminaries
440(6)
10.2 Linear Representation and N((a†a)n)
446(3)
10.3 Wick's Theorem and q-Normal Ordering
449(3)
10.4 p-Normal Ordering
452(5)
10.5 Noncrossing Normal Ordering
457(12)
10.5.1 Some Preliminary Observations
459(1)
10.5.2 Noncrossing Normal Ordering of (ar(a†)s)n
460(2)
10.5.3 Noncrossing Normal Ordering of (ar+(a†)s)n
462(3)
10.5.4 k-Ary Trees and Lattice Paths
465(4)
10.6 Exercises
469(2)
10.7 Research Directions and Open Problems
471(2)
A Solutions and Hints 473(28)
B Identities 501(2)
C Power Series and Binomial Theorem 503(4)
D Chebychev Polynomials of the Second Kind 507(4)
E Linear Algebra and Algebra Review 511(2)
F Complex Analysis Review 513(4)
G Coherent States 517(2)
H C++ Programming 519(18)
I Tables 537(6)
J Notation 543(4)
Bibliography 547(26)
Index 573
Toufik Mansour is a professor in the Department of Mathematics at the University of Haifa. Dr. Mansour has authored/co-authored more than 200 papers and is a reviewer for many journals, including Advances in Applied Mathematics, Discrete Mathematics, Discrete Applied Mathematics, European Journal of Combinatorics, and the Journal of Combinatorial Theory Series A. His research focuses on pattern avoidance in permutations, colored permutations, set partitions, words, and compositions.