Counting with Symmetric Functions

by Jeffrey Remmel
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Jeffrey Remmel Counting with Symmetric Functions
Jeffrey Remmel - Counting with Symmetric Functions

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This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics.  It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas.

The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions.  Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions.  Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4.  The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya’s enumeration theorem using symmetric functions.  Chapters 7 and 8 are more specialized than the preceding ones, covering consecutive pattern matches in permutations, words, cycles, and alternating permutations and introducing the reciprocity method as a way to define ring homomorphisms with desirable properties.

Counting with Symmetric Functions will appeal to graduate students and researchers in mathematics or related subjects who are interested in counting methods, generating functions, or symmetric functions.  The unique approach taken and results and exercises explored by the authors make it an important contribution to the mathematical literature.


Jeffrey Remmel
Anthony Mendes

Further information

Illustrations Note:
X, 292 p. 209 illus.
Table of Contents:

Preface.- Permutations, Partitions, and Power Series.- Symmetric Functions.- Counting with the Elementary and Homogeneous.- Counting with a Nonstandard Basis.- Counting with RSK.- Counting Problems that Involve Symmetry.- Consecutive Patterns.- The Reciprocity Method.- Appendix: Transition Matrices.- References.- Index.

A self-contained introduction to symmetric functions and their use in counting problems

First book to consider many of the methods and results presented

Unifies a large number of results? in the theory of permutation enumeration

Numerous exercises with full solutions included throughout

Includes supplementary material:

Media Type:
Springer International Publishing

"This book provides a current survey of techniques and applications of symmetric functions to enumeration theory, with emphasis on the combinatorics of the transition matrices between bases. ... Each chapter ends with a substantial number of exercises along with full solutions, as well as accurate bibliographic notes. The book is definitely a very interesting addition to the literature on the subject." (Domenico Senato, Mathematical Reviews, February, 2017)

"Though the authors target graduate students, advanced undergraduates will also surely have the necessary prerequisites, easily grasp the book's goals, and find many chapters accessible. ... Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners." (D. V. Feldman, Choice, Vol. 53 (12), September, 2016)

1st ed. 2015
Number of Pages:

Master Data

Product Type:
Hardback book
Package Dimensions:
0.243 x 0.165 x 0.024 m; 0.598 kg
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