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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 Polya'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.
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 Polya'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.
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