Counting with symmetric functions / by Jeffrey Remmel, Anthony Mendes.

Format:

Book

Author/Creator:

Remmel, Jeffrey., Author.

Mendes, Anthony, Author.

Series:

Developments in Mathematics, 1389-2177 ; 43

Language:

English

Subjects (All):

Combinatorial analysis.

Functions, Special.

Sequences (Mathematics).

Combinatorics.

Special Functions.

Sequences, Series, Summability.

Local Subjects:

Combinatorics.

Special Functions.

Sequences, Series, Summability.

Physical Description:

1 online resource (297 p.)

Edition:

1st ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2015.

Language Note:

English

Summary:

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.

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.

Notes:

Description based upon print version of record

Includes bibliographical references and index.

ISBN:

3-319-23618-0