Math 155 (Lecture 1)
August 30, 2011
This course is meant to serve as an introduction to the study of combinatorics. We might therefore
begin by asking the question: what is combinatorics? According to Wikipedia, combinatorics is a branch
of mathematics c
Math 155 (Lecture 6)
September 13, 2011
In this lecture, we will continue our discussion of natural operations on species.
Denition 1. Let S and T be species. We dene a new species ST , called the product of S and T , as
follows:
(a) Let I be a nite set.
Math 155 (Lecture 7)
September 16, 2011
In this lecture, we describe some applications of the composition formula
FS T (x) = FS (FT (x)
proved in the last lecture. We will be particularly interested in the case where S = SSet is the species of
nite sets,
Math 155 (Lecture 5)
September 13, 2011
Our goal in this lecture is to introduce Joyals theory of species, which provides a convenient language
for formalizing many ideas in enumerative combinatorics.
Denition 1. A species is a rule S which does the follo
Math 155 (Lecture 4)
September 8, 2011
Let us continue our analysis of the problem posed at the end of the previous lecture. For each n 0,
we let Dn denote the number of derangements of the set cfw_1, . . . , n. In the last lecture, we determined the
expo
Math 155 (Lecture 2)
September 1, 2011
The goal of this lecture is to introduce a basic technique in combinatorics: the method of generating
functions. Let begin with a simple counting problem.
Question 1. How many ways are there to tile a 2-by-n board wi
Math 155 (Lecture 3)
September 8, 2011
In this lecture, well consider the answer to one of the most basic counting problems in combinatorics.
Question 1. How many ways are there to choose a k -element subset of the set cfw_1, 2, . . . , n?
The answer to t
Math 155 (Lecture 8)
September 19, 2011
Denition 1. Let G be a graph. A path in G is a sequence of vertices v0 , . . . , vn such that vi is adjacent to
vi1 for 1 i n. In this case, we also say (v0 , . . . , vn ) is a path from v0 to vn . We say that a pat
Math 155 (Lecture 9)
September 21, 2011
In the last few lectures, we have discussed Joyals theory of species. This is a convenient language for
analyzing labelled counting problems: that is, for answering questions of the following form:
Proto-Question 1.
Math 155 (Lecture 14)
October 5, 2011
In the last few lectures, we have discussed the cycle index of a nite group G acting on a set X . The
cycle index contains a lot of enumerative information in a conveniently accessible form. However, it is not
always
Math 155 (Lecture 15)
October 10, 2011
Recall that species S is said to be molecular if there is exactly one S -structure, up to isomorphism.
Equivalently, S is molecular if there exists an integer n such that S [ m ] is empty for m = n, and S [ n ] is
ac
Math 155 (Lecture 13)
September 30, 2011
In this lecture, we will study some of the formal properties of the cycle index ZG of a nite group G acting
on a set X . Ideally, we would like to have some recipe for computing the cycle index of X in terms of the
Math 155 (Lecture 12)
September 28, 2011
In this lecture we will work out some more examples of cycle indices and applications of Polyas theorem.
Question 1. Up to rotational symmetry, how many ways can we color the faces of a cube using a set of
colors T
Math 155 (Lecture 10)
September 22, 2011
In the last lecture, we proved the following result:
Theorem 1 (Polya Enumeration Theorem). Let G be a nite group, X a nite G-set, and T a nite set
1
with t elements. Then the quotient G\T X has cardinality |G| gG
Math 155 (Lecture 11)
September 25, 2011
In the last lecture, we explained how to use Polyas theorem to count the number if isomorphism classes
of graphs of size n. The answer was given by the sum
1
n!
2o() ,
n
where o( ) denotes the number of orbits of