MATH 681
1
Notes
Combinatorics and Graph Theory I
Graph Theory
1.1
More on edge-counting
The number of edges incident on a vertex will be relevant for a number of reasons.
Denition 1. The degree of a vertex v in a graph G, denoted dG (v ), is the number o
MATH 681
1
Notes
Combinatorics and Graph Theory I
The Inclusion-Exclusion Principle
Our next step in developing the twelvefold way will deal with the surjective functions. Well build
these through the use of inclusion-exclusion.
In its most basic form, in
MATH 681
1
Notes
Combinatorics and Graph Theory I
What is combinatorics?
Combinatorics is the branch of mathematics dealing with things that are discrete, such as the
integers, or words created from an alphabet. This is in contrast to analysis, which deal
MATH 681
Final Exam
Answer exactly four of the following six questions. Indicate which four you would like
graded!
Binomial coecients, Stirling numbers, and arithmetic expressions need not be simplied
in your answers.
1. (15 points) Answer the following q
MATH 681
Final Exam
Answer exactly four of the following six questions. Indicate which four you would like
graded!
Binomial coecients, Stirling numbers, and arithmetic expressions need not be simplied
in your answers.
1. (12 students attempted this proble
MATH 681
Exam #2
Answer exactly four of the following six questions. Indicate which four you would like
graded!
Binomial coecients, Stirling numbers, and arithmetic expressions need not be simplied
in your answers.
1. (10 points) A necklace consists of 6
MATH 681
Exam #2
1. (10 of 12 students attempted this) A necklace consists of 6 gems; the gems can
be garnets, tourmaline, or zircons. Two necklaces are considered to be identical if one
can be obtained by rotating or ipping the other.
(a) (5 points) Find
MATH 681
Exam #1
Answer exactly four of the following six questions. Indicate which four you would like
graded!
Binomial coecients, Stirling numbers, and arithmetic expressions need not be simplied
in your answers.
1. (10 points) Prove the combinatorial i
MATH 681
Exam #1
1. (8 of 12 students attempted this) Prove the combinatorial identity
n2n1 + n(n 1)2n2 . You may use any method you like.
n
k=1
k2
n
k
=
A simple approach is purely combinatorial: the left side of the equation represents
the number of way
MATH 681
1
Notes
Combinatorics and Graph Theory I
Fun with the combination statistic
Of the 12 enumerative functions weve seen, the one with the most obvious utility is the combination
statistic, a.k.a. the binomial coecient:
n
k
n!
n(n 1)(n 2) . . . (n k
MATH 681
1
1.1
Notes
Combinatorics and Graph Theory I
Generating functions, continued
Generating functions and partitions
We can make use of generating functions to answer some questions a bit more restrictive than weve
done so far:
Question 1: Find a gen
MATH 681
1
Notes
Combinatorics and Graph Theory I
Graph Theory
On to the good stu! Posets are but one kind of combinatorial structure. A somewhat more
complicated structure is the graph, which describes relationships among a set of nodes. In
applciation,
MATH 681
1
1.1
Notes
Combinatorics and Graph Theory I
Chains and Antichains
Maximality and maximum-, uh, -ness?
To review, the denitions of a chain and antichain:
Denition 1. A chain is a totally ordered subset of a poset S ; an antichain is a subset of a
MATH 681
1
1.1
Notes
Combinatorics and Graph Theory I
Posets
Extreme elements
Last week we dened maximal, minimal, greatest, and least elements of a poset. We will explicitly
determine the useful properties of these elements.
The proofs below will general
MATH 681
1
Notes
Combinatorics and Graph Theory I
Restricted symmetry-enumerations
So far weve built what is essentially a magic bullet to answer the question: if we freely paint an
object with k colors, how many distinct colorings are there subject to sy
MATH 681
1
Notes
Combinatorics and Graph Theory I
Equivalence classes of symmetries
This denition is probably familiar, but its useful for discussing classication under symmetry.
Denition 1. A set R of ordered pairs from S S is called a relation on S ; tw
MATH 681
1
Notes
Combinatorics and Graph Theory I
Catalan numbers
Previously, we used generating functions to discover the closed form Cn = 2
actually turn out to be marvelously simpliable:
1/2
n+1
(4)n . This will
1/2
(4)n
n+1
( 1 )( 1 )( 3 ) ( 12n )
2
2
MATH 681
1
Notes
Combinatorics and Graph Theory I
Recurrence relations, continued yet again
One last lousy class of recurrences we should be able to solve:
1.1
Systems of recurrence relations
Sometimes multiple recurrences working in tandem are more eecti
MATH 681
1
Notes
Combinatorics and Graph Theory I
Recurrence relations, continued continued
Linear homogeneous recurrences are only one of several possible ways to describe a sequence as a
recurrence. Here are several other situations which may arise.
1.1
MATH 681
1
1.1
Notes
Combinatorics and Graph Theory I
Generating functions, continued
Exponential generating functions and set-partitions
At this point, weve come up with good generating-function discussions based on 3 of the 4 rows
of our twelvefold way.
COMBINATORICS MASTERS EXAM
Instructions: Work out as many problems as you can. Focus on complete and correct solutions, because partial credit will only be sparingly
awarded, and complete solutions are required to pass. Binomial coecients, Stirling number
TAKE HOME PORTION OF THE FINAL EXAM, DUE ON
EXAM DAY
(1) Let Rd be the (innite) poset, whose ground set is the set of d-tuples of
real numbers, and (a1 , . . . , ad ) (b1 , . . . , bd ) if and only if ai bi for all
i = 1, . . . , d. Prove that the dimensi
MATH 681
Problem Set #5
This problem set is due at the beginning of class on November 10.
2
1. (5 points) Find an asymptotically accurate approximation for n in terms of polyn
nomials, exponentials, and self-exponentials. You may write it in big-O notatio
MATH 681
Problem Set #5
This problem set is due at the beginning of class on November 10.
2
1. (5 points) Find an asymptotically accurate approximation for n in terms of polyn
nomials, exponentials, and self-exponentials. You may write it in big-O notatio
MATH 681
Problem Set #4
This problem set is due at the beginning of class on October 22.
1. (10 points) For this problem, it will be helpful to note the following two power series
expansions:
x2 x4 x6
ex + ex
=1+
+
+
+
2
2!
4!
6!
ex ex
x3 x5 x7
=x+
+
+
+
MATH 681
Problem Set #4
1. (10 points) For this problem, it will be helpful to note the following two power series
expansions:
ex + ex
x2 x4 x6
=1+
+
+
+
2
2!
4!
6!
ex ex
x3 x5 x7
=x+
+
+
+
2
3!
5!
7!
(a) (5 points) Find an exponential generating functi
MATH 681
Problem Set #3
This problem set is due at the beginning of class on October 1.
1. (10 points) Prove the following combinatorial identities:
(a) (5 points) Recall that S (n, k ) is equal to the number of ways to subdivide an
n-element set into k n
MATH 681
Problem Set #3
1. (10 points) Prove the following combinatorial identities:
(a) (5 points) Recall that S (n, k ) is equal to the number of ways to subdivide an
n-element set into k nonempty parts. Produce a combinatorial argument to show
that S (
MATH 681
Problem Set #2
Show work for each problem. Answers without justication, or justied solely by direct
enumeration, are not acceptable. Arithmetic expressions may be left unsimplied.
This problem set is due at the beginning of class on September 17.