HOMEWORK 1
(1) Let S be a set of integers cfw_s0 , s1 , . . . , sk and assume that s0 > s1 > >
k
sk . Let M (S ) = i=0 (1)i si . So M (cfw_1, 2, 5, 6, 9) = 7, M (cfw_3) = 3, and
M () = 0. Compute
M (
HOMEWORK #5, DUE ON 11/08
(1) The goal of this problem to formally introduce generating functions. Each
student was assigned one part of this problem. If you dont know your
assignment, please let me k
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.
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.
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
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 a
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
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
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
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 obtain
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
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 re
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 disc
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
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 describ
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 orde
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 prop
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 colo
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 o
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
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
Sometime
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
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, a
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, a
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
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!
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 th
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 p
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 proble