Advances in Combinatorial Methods and Applications to Probability and Statistics

575 Pages·1996·11.397 MB·English

by  Ira M. Gessel, Sangwook Ree (auth.), N. Balakrishnan (eds.)

Statistics for Industry and Technology
Series Editor
N. Balakrishnan
McMaster University
Editorial Advisory Board
Max Engelhardt
EG&G Idaho, Inc.
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Kazuyuki Suzuki
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University of Electro Communications
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Tokyo 182
Japan
In Honor of
Sri Gopal Mohanty
SRI GOPAL MOHANTY
Advances in Combinatorial
Methods and Applications
to Probability and Statistics
N. Balakrishnan
Editor
1997
Birkhauser
Boston • Basel • Berlin
N. Balakrishnan
Department of Mathematics
and Statistics
McMaster University
Hamilton, Ontario L8S 4Kl
Canada
Library of Congress Cataloging-in-Publication Data
Advances in combinatorial methods and applications to probability and statistics / N.
Balakrishnan, editor.
p. cm. --(Statistics for industry and technology)
Includes bibliographical references and index.
ISBN-13: 978-1-4612-8671-4 e-ISBN-13: 978-1-4612-4140-9
DOl: 10.107/978-1-4612-4140-9
paper)
I. Combinatorial probabilities. 2. Mathematical statistics.
I. Balakrishnan, N., 1956- II. Series.
QA273.45.A38 1997
519.2--DC21 97-6185
CIP
m®
Printed on acid-free paper
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ISBN-13: 978-1-4612-8671-4
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9 8 7 6 543 2 I
Contents
Preface xvii
Sri Gopal Mohanty-Life and Works xix
List of Contributors xxvii
List of Tables xxxi
List of Figures xxxiii
PART I-LATTICE PATHS AND COMBINATORIAL METHODS
1 Lattice Paths and Faber Polynomials
Ira M. Gessel and Sangwook Ree 3
1.1 Introduction, 3
1.2 Faber Polynomials, 6
1.3 Counting Paths, 7
1.4 A Positivity Result, 10
1.5 Examples, 11
References, 13
2 Lattice Path Enumeration and U mbral Calculus
Heinrich Niederhausen 15
2.1 Introduction, 15
2.1.1 Notation, 16
2.2 Initial Value Problems, 16
2.2.1 The role of eX, 18
2.2.2 Piecewise affine boundaries, 18
2.2.3 Applications: Bounded paths, 19
2.3 Systems of Operator Equations, 20
2.3.1 Applications: Lattice paths with
several step directions, 21
2.4 Symmetric Sheffer Sequences, 21
2.4.1 Applications: Weighted left turns, 22
2.4.2 Paths inside a band, 23
Vlll Contents
2.5 Geometric Sheffer Sequences, 24
2.5.1 Applications: Crossings, 25
References, 26
3 The Enumeration of Lattice Paths With Respect
to Their Number of Turns
C. K rattenthaler 29
3.1 Introduction, 29
3.2 Notation, 31
3.3 Motivating Examples, 31
3.4 Turn Enumeration of (Single) Lattice Paths, 36
3.5 Applications, 44
3.6 Nonintersecting Lattice Paths and Turns, 47
References, 55
4 Lattice Path Counting, Simple Random Walk
Statistics, and Randomizations: An Analytic
Approach
Wolfgang Panny and Walter Katzenbeisser 59
4.1 Introduction, 59
4.2 Lattice Paths, 60
4.3 Simple Random Walks, 64
4.4 Randomized Random Walks, 70
References, 74
5 Combinatorial Identities: A Generalization of
Dougall's Identity
Erik Sparre Andersen and Mogens Esram Larsen 77
5.1 Introduction, 77
5.2 The Generalized Pfaff-Saalschiitz Formula, 80
5.3 A Modified Pfaff-Saalschiitz Sum of Type
11(4,4, 1)N, 82
5.4 A Well-Balanced IJ(5, 5, 1)N Identity, 83
5.5 A Generalization of Dougall's Well-Balanced
11(7,7, 1)N Identity, 85
References, 87
6 A Comparison of Two Methods for Random
Labelling of Balls by Vectors of Integers
Doran Zeilberger 89
6.1 First Way, 89
6.2 Second Way, 89
6.3 Variance and Standard Deviation, 91
Contents ix
6.4 Analysis of the Second Way, 92
References, 93
PART II-ApPLICATIONS TO PROBABILITY PROBLEMS
7 On the Ballot Theorems
Lajos Takacs 97
7.1 Introduction, 97
7.2 The Classical Ballot Theorem, 97
7.3 The Original Proofs of Theorem 7.2.1, 100
7.4 Historical Background, 102
7.5 The General Ballot Theorem, 104
7.6 Some Combinatorial Identities, 107
7.7 Another Extension of The Classical Ballot
Theorem, 109
References, 111
8 Some Results for Two-Dimensional Random Walk
Endre Csaki 115
8.1 Introduction, 115
8.2 Identities and Distributions, 118
8.3 Pairs of LRW Paths, 120
References, 123
9 Random Walks on 8L(2, F2) and Jacobi Symbols of
Quadratic Residues
Toshihiro Watanabe 125
9.1 Introduction, 125
9.2 Preliminaries, 126
9.3 A Calculation of the Character X(O'M,m) and
Its Relation, 129
References, 133
10 Rank Order Statistics Related to a Generalized
Random Walk
Jagdish Saran and Sarita Rani 135
10.1 Introduction, 135
10.2 Some Auxiliary Results, 136
10.3 The Technique, 138
10.4 Definitions of Rank Order Statistics, 139
10.5 Distributions of N:,~,(a) and Rt,~,(a), 140
10.6 Distributions of At,n(a) and Rf':,n(a), 144
10.7 Distributions of N;,n(a) and R~,r,,(a), 148
References, 151
x Contents
11 On a Subset Sum Algorithm and Its Probabilistic
and Other Applications
V. G. Voinov and M. S. Nikulin 153
11.1 Introduction, 153
11.2 A Derivation of the Algorithm, 154
11.3 A Class of Discrete Probability Distributions, 159
11.4 A Remark on a Summation Procedure When
Constructing Partitions, 160
References, 162
12 I and J Polynomials in a Potpourri of Probability
Problems
Milton Sobel 165
12.1 Introduction, 165
12.2 Guide to the Problems of this Paper, 166
12.3 Triangular Network with Common Failure
Probability q for Each Unit, 171
12.4 Duality Levels in a Square with Diagonals
That Do Not Intersect: Problem 12.5, 177
References, 183
13 Stirling Numbers and Records
N. Balakrishnan and V. B. Nevzorov 189
13.1 Stirling Numbers, 189
13.2 Generalized Stirling Numbers, 190
13.3 Stirling Numbers and Records, 193
13.4 Generalized Stirling Numbers and Records
in the FO<-scheme, 195
13.5 Record Values from Discrete Distributions
and Generalized Stirling Numbers, 197
References, 198
PART III-ApPLICATIONS TO URN MODELS
14 Advances in Urn Models During The
Past Two Decades
Samuel Kotz and N. Balakrishnan 203
14.1 Introduction, 203
14.2 P6lya-Eggenberger Urns and Their Generalizations
and Modifications, 206
14.3 Generalizations of the Classical Occupancy
Model, 216
14.4 Ehrenfest Urn Model, 219