This work assumes on the part of the reader a knowledge of modern enumerative combina-
torics as can be found, e.g., in [63].
In sections 5.2, 5.3, 6.2, 7.4, 7.5 and 7.7 the theory of symmetric functions is used. For
this theory we follow the terminology and notations of [38], Chapter I.
We collect here some notations that will be adhered to throughout this work.
Z the integers
N the nonnegative integers
P the positive integers
Q, R, C the rational, real and complex numbers
[p] the set {1,2,...,p} , ( p P )
[a, b] the set {x 6 R, a x &}, (a, b 6 R)
[aJ the largest integer a, (a £ R)
\a] the smallest integer a, (a R)
\S\ the cardinality of the set S
l±) disjoint union of sets
A[xi,..., xn] the ring of polynomials in the variables x i , . . . , xn with coefficients in the
integral domain A
A(xi,..., xn) the ring of rational functions in the variables a?i,..., xn with coefficients in
the integral domain A
A[[zi,..., xn]] the ring of formal power series in the variables xi,..., xn with coefficients
in the integral domain A
"f ' = J,
Each result (theorem, corollary, proposition, or lemma) is numbered consecutively. So, for
example, Theorem 2.3.3 is the third result in the third section of chapter 2 (i.e. in section
2.3). We denote with a the end of a proof. A appearing at the end of the statement of
a result signifies that the result should be obvious at that stage of reading. A result whose
statement is neither followed by a proof nor by a is a result whose proof goes beyond the
scope of this work. A reference to a proof will always be given in such cases.
Sij the Kronecker delta:6ij
def / 1
\o if,
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