Foreword

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

i*

*

=

\o if,

VI