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Article
Connection Coefficients Between Generalized Rising and Falling Factorial Bases
Let \({\mathcal{S} = (s_1, s_2, \ldots)}\) S ...
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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.
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Article
Row-Strict Quasisymmetric Schur Functions
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in m...
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Article
Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm
We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S n ...
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Article
Generating Functions for Permutations Avoiding a Consecutive Pattern
Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We h...
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Article
Open AccessSymmetric Functions and Generating Functions for Descents and Major Indices in Compositions
In [18], Mendes and Remmel showed how Gessel’s generating function for the distributions of the number of descents, the major index, and the number of inversions of permutations in the symmetric group could be...
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Article
Classifying Descents According to Parity
In this paper we refine the well-known permutation statistic “descent” by fixing parity of (exactly) one of the descent’s numbers. We provide explicit formulas for the distribution of these (four) new statisti...