Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.
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'… thorough and self-contained … presentation of … topics is very well organised … provides an ample amount of examples and illustrations, as well as a comprehensive bibliography. It is valuable both as a reference work for researchers working in the field and as an accessible introduction suitable for students at an advanced graduate level.' EMS Newsletter
Book Description
The definitive treatment of analytic combinatorics. This self-contained text covers the mathematics underlying the analysis of discrete structures, with thorough treatment of a large number of applications. Exercises, examples, appendices and notes aid understanding: ideal for individual self-study or for advanced undergraduate or graduate courses.
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4.7 out of 5 stars
3 reviews
4 of 4 people found the following review helpful
5.0 out of 5 stars
the bible of the subject
7 Mar 2011
By Narada
- Published on Amazon.com
Format:Hardcover|Amazon Verified Purchase
Flajolet and his group at INRIA (B. Salvy, P. Zimmermann, many others) have created the field of "analytic combinatorics", and this book has everything you might possibly want to know about the subject (since the selection of topics is very thoughtful, it does not have that much you would NOT want to know). The only reason to possibly not get this book (assuming you are interested in the subject) is that the PDF is available free on Flajolet's web page, but the book for is quite convenient and quite reasonably priced.
1 of 1 people found the following review helpful
5.0 out of 5 stars
Totally Awesome
6 May 2013
By ralph kelsey
- Published on Amazon.com
Format:Hardcover|Amazon Verified Purchase
This book covers a LOT of ground FAST. You can think of it as the third volume in the following sequence:
(1) Algorithms; Sedgewick and Wayne. Appropriate for an upper level undergrad/grad text book for a university algorithms course.
(2) Analysis of Algorithms; Sedgewick and Flajolet. Much more detailed treatment of algorithms, and using the Generating Function approach to Algorithm Analysis.
(3) Analytic Combinatorics: S&J. Advanced aspects of the above.
Here is the setup: For some category of computer science or combinatorical object, let A[n] denote how many distinct examples there are of size n. Then the complex function f(z) = sum(A[n] z^n) is the "Generating function" for A[n].
The S&J methodology is (1) generate an equation for f(z) directly from the combinatorical object. (2) Use this equation to solve for f(z), or at least obtain info about it. (3) Use classical complex analysis (singularity analysis and saddle point methods) to obtain excellent estimates of A[n].
I am confident this will get easy once I work out about 100 examples!
(1) Algorithms; Sedgewick and Wayne. Appropriate for an upper level undergrad/grad text book for a university algorithms course.
(2) Analysis of Algorithms; Sedgewick and Flajolet. Much more detailed treatment of algorithms, and using the Generating Function approach to Algorithm Analysis.
(3) Analytic Combinatorics: S&J. Advanced aspects of the above.
Here is the setup: For some category of computer science or combinatorical object, let A[n] denote how many distinct examples there are of size n. Then the complex function f(z) = sum(A[n] z^n) is the "Generating function" for A[n].
The S&J methodology is (1) generate an equation for f(z) directly from the combinatorical object. (2) Use this equation to solve for f(z), or at least obtain info about it. (3) Use classical complex analysis (singularity analysis and saddle point methods) to obtain excellent estimates of A[n].
I am confident this will get easy once I work out about 100 examples!
7 of 10 people found the following review helpful
4.0 out of 5 stars
maybe a little long and hard to read but excellent!
4 Dec 2009
By R. Bagula
- Published on Amazon.com
Format:Hardcover
Sort of the hidden underbelly of combinatorial mathematics is the relationship to
fractals and this book actually seems to admit it exists.
Some of the different types of generating function are hard to follow
and a reference to Roman's Umbral calculus or Sheffer sequences
would be useful as well, but a lot of very useful information
is here and they do mention Dr. Sloane's EIS.
Not since I got a look at the classic Riordan text have I seen
a book that packs this kind of impact on the field
of combinatorial mathematics.
This book is one for students who are serious about learning
the developing science of combinatorial analysis.
I just wish I could afford it
and not have to check it out of the library to be able to read it!
The connection of Bernoulli functions to bosons
and Euler functions to leptons isn't touched on,
but it is the reason that combinatorial analysis is important to physics.
So you can't give the book 5 stars even as good as what is presented is.
fractals and this book actually seems to admit it exists.
Some of the different types of generating function are hard to follow
and a reference to Roman's Umbral calculus or Sheffer sequences
would be useful as well, but a lot of very useful information
is here and they do mention Dr. Sloane's EIS.
Not since I got a look at the classic Riordan text have I seen
a book that packs this kind of impact on the field
of combinatorial mathematics.
This book is one for students who are serious about learning
the developing science of combinatorial analysis.
I just wish I could afford it
and not have to check it out of the library to be able to read it!
The connection of Bernoulli functions to bosons
and Euler functions to leptons isn't touched on,
but it is the reason that combinatorial analysis is important to physics.
So you can't give the book 5 stars even as good as what is presented is.
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4.7 out of 5 stars
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