The ever-growing applications and richness of approaches to the Riordan group is captured in this comprehensive monograph, authored by those who are among the founders and foremost world experts in this field. The concept of a Riordan array has played a unifying role in enumerative combinatorics over the last three decades. The Riordan arrays and Riordan group is a new growth point in mathematics that is both being influenced by, and continuing its contributions to, other fields such as Lie groups, elliptic curves, orthogonal polynomials, spline functions, networks, sequences and series, Beal conjecture, Riemann hypothesis, to name several. In recent years the Riordan group has made links to quantum field theory and has become a useful tool for computer science and computational chemistry. We can look forward to discovering further applications to unexpected areas of research. Providing a baseline and springboard to further developments and study, this book may also serve as a text for anyone interested in discrete mathematics, including combinatorics, number theory, matrix theory, graph theory, and algebra.
Arvustused
Each chapter ends with a set of exercises, designed to help the reader to develop a deeper understanding of the theory and methods of Riordan arrays, and a reference section. Solutions to selected exercises are also provided at the end of the book. It is a pleasure to read such a comprehensive and clearly written book on Riordan arrays and the Riordan group. This book excellently introduces and develops a fascinating new research area of enumerative combiatronics. (Chen Sheng, zbMATH 1498.05002, 2022)
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1 | (18) |
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1.1 What are Riordan Arrays? |
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1 | (2) |
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1.2 Origins and Motivation |
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3 | (6) |
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1.3 Elementary Applications |
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9 | (10) |
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16 | (1) |
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17 | (2) |
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2 Extraction of Coefficients and Generating Functions |
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19 | (28) |
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19 | (2) |
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2.2 Coefficient Extraction |
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21 | (3) |
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2.3 Lagrange Inversion Theorem |
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24 | (3) |
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27 | (20) |
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43 | (3) |
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46 | (1) |
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47 | (22) |
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3.1 Riordan Arrays and the Riordan Group |
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47 | (11) |
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3.2 Some Special Subgroups |
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58 | (4) |
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3.3 Several Aspects of the Riordan Group |
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62 | (7) |
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66 | (1) |
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66 | (3) |
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4 Characterization of Riordan Arrays by Special Sequences |
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69 | (32) |
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4.1 The A- and Z- Sequences |
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70 | (7) |
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77 | (13) |
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4.3 Is It a Riordan Array? |
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90 | (11) |
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97 | (1) |
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98 | (3) |
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5 Combinatorial Sums and Inversions |
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101 | (22) |
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101 | (8) |
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5.2 Combinatorial Inversions |
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109 | (14) |
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120 | (1) |
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121 | (2) |
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6 Generalized Riordan Arrays |
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123 | (90) |
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6.1 Exponential Riordan Arrays |
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124 | (6) |
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6.2 Generalized Riordan Arrays and the Riordan Group |
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130 | (8) |
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6.3 Relations Between Riordan Arrays and Sheffer Sequences |
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138 | (14) |
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6.4 Special Riordan Arrays and Sheffer Sequences |
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152 | (30) |
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6.5 Double Riordan Arrays and Sheffer Polynomial Pairs |
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182 | (31) |
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205 | (4) |
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209 | (4) |
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7 Extensions of the Riordan Group |
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213 | (30) |
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7.1 Three-Dimensional Riordan Group |
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213 | (4) |
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7.2 Three-Dimensional Riordan Arrays |
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217 | (13) |
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7.3 The Riordan Group in Several Variables |
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230 | (13) |
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238 | (2) |
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240 | (3) |
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8 Q-Analogs of Riordan Arrays |
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243 | (16) |
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8.1 Combinatorial q-Analogs |
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243 | (2) |
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8.2 Eulerian Generating Functions |
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245 | (1) |
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246 | (5) |
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8.4 Combinatorial Applications of the Q-Riordan Arrays |
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251 | (8) |
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255 | (2) |
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257 | (2) |
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259 | (76) |
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9.1 Orthogonal Polynomials and Riordan Arrays |
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260 | (18) |
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9.2 Exponential Riordan Arrays and Classical Orthogonal Polynomials |
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278 | (7) |
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9.3 Orthogonal Polynomials as Moments |
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285 | (18) |
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9.4 Combinatorial Polynomials as Moments of Riordan Arrays |
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303 | (13) |
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9.5 Continued Fractions and Riordan Arrays |
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316 | (19) |
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332 | (1) |
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333 | (2) |
| Solutions |
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335 | (24) |
| Index |
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359 | |
Louis Shapiro is a Professor Emeritus at Howard University. He has taught at Howard for 56 years with research interests in enumerative combinatorics, finite groups, and of course the Riordan group. He is also an avid runner and square dance caller.
Renzo Sprugnoli is a mathematician and computer scientist working in the fields of analysis of algorithms and combinatorics. Of particular note his works on the computation of combinatorial sums and the enumeration of lattice paths through Riordan arrays and the special sequences arising in this context. As a full professor, he has taught classes on algorithms, data structures and databases at the University of Florence.
Paul Barry is Emeritus professor of mathematics at Waterford Institute of Technology, Ireland. He carries out research into integer sequences and Riordan arrays. He is the author of the book Riordan Arrays: A Primer.
Gi-Sang Cheon is a mathematician working in the field of combinatorial matrix theory. He is a professor at the Department of Mathematics of Sungkyunkwan University in South Korea.
Tian-Xiao He is a professor of mathematics and the Earl and Marian A. Beling Professor of Natural Science at Illinois Wesleyan University. His research fields include enumerative combinatorics, Riordan group, numerical analysis, approximate theory, wavelet analysis, and number theory.
Donatella Merlini is a computer scientist working in the fields of analysis of algorithms, enumerative combinatorics, symbolic computation and data mining, subjects she teaches at the University of Florence. Since her PhD thesis, she has studied both the theoretical aspects of Riordan arrays and the applications in the context of algorithms and data structures analysis.
Weiping Wang is an associate professor of School of Science at ZhejiangSci-Tech University, China. His research fields include enumerative combinatorics, combinatorial algorithms, and special functions. His research topics are related to combinatorial sequences, combinatorial summations, and multiple zeta values.
