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Book
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Chapter
Symmetric Functions
The ring of symmetric functions is introduced. The six standard bases for symmetric functions; namely, the monomial, elementary, homogeneous, power, forgotten, and Schur symmetric functions, are defined. Numer...
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Chapter
Counting with Nonstandard Bases
Generalizing the relationship between the elementary and power symmetric functions, we define a new basis for the ring of symmetric functions which has an expansion in terms of specially weighted brick tabloid...
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Chapter
Counting Problems That Involve Symmetry
Symmetric functions are used to prove Pólya’s enumeration theorem, allowing us to count objects modulo symmetries.
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Chapter
The Reciprocity Method
In previous chapters, we defined ring homomorphisms φ \(\varphi\) ...
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Chapter
Permutations, Partitions, and Power Series
Statistics on permutations and rearrangements are defined and relationships between q-analogues of n, n ...
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Chapter
Counting with the Elementary and Homogeneous Symmetric Functions
The relationship between the elementary and homogeneous symmetric functions, specifically the expansion involving brick tabloids, is used to find an assortment of generating functions. We are able to count and...
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Chapter
Counting with RSK
The RSK algorithm is introduced and used to find generating functions for permutation statistics. Connections are made to increasing subsequences in permutations and words and the Schur symmetric functions. A q-a...
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Chapter
Consecutive Patterns
This chapter applies the machinery of ring homomorphisms on symmetric functions to understand consecutive pattern matches in permutations, words, cycles, and in alternating permutations.