Homework assignment Number 1
Math 145 / due on 01/12/2007
1. Problems Section 1.8: 8, 21, 24, 27, 28, 34. 2. Problem 1.8.5 page 21. 3. How many permutations of the set S = {1, . . . , n} have a single cycle? How many l cycles can you form on S? 4. A
MATH 145: HOMEWORK 1
ANDREW BERGET
Do not hand in the [bracketed] problems, they are simply suggestions of what
might be a good problem for you to practice with. Once you feel comfortable with
these you should be able to do the unbracketed problems.
These
MATH 145: HOMEWORK 2
ANDREW BERGET
As before, do not hand in [bracketed] problems. This homework is due on Wednesday January 20.
Problem A. We toss a fair coin n times and get get h heads and t tails where h > t are xed integers.
Your goal in this problem
MATH 145: HOMEWORK 3
ANDREW BERGET
This homework is due on Wednesday January 27. In these problem you will need to use the fact that
a
.
axn =
1x
n0
Read section 5.3 and solve problems 5.23(a) and 5.25+.
Problem A. In this problem we consider the generati
MATH 145: HOMEWORK 4
ANDREW BERGET
This is due Friday February 5!
Problem A. When I write Determine the generating function, I mean write the generating function
in a closed, simple form.
(1) [Determine the generating function of the constant sequence an
MATH 145: HOMEWORK 4
ANDREW BERGET
This is due Friday February 19.
Problem A. Is there a simple graph on 5 vertices cfw_v1 , v2 , v3 , v4 , v5 with deg(vi ) = i. Do not do this by
attempting to draw every graph on 5 vertices.
Problem B. The complement of
MATH 145: HOMEWORK 6
ANDREW BERGET
Solve problems 10.21, 10.22, 10.23, 10.28 (this is asking for unlabeled tress!),
10.30+, 10.31, 10.34+.
Problem A.+ Let G be a graph and T is a spanning tree of G. Let f be an
edge of G which is not an edge of T .
(1) Pr
MATH 145: HOMEWORK 7
ANDREW BERGET
Problem A. Prove the Cayley-Prfer theorem:
u
d (1)1 dT (2)1
x2
(x1 + x2 + + xn )n2 =
x1T
. . . xdT (n)1 .
n
T
Here the sum is over labeled trees T with vertex set [n] and dT (i) is the degree of
vertex i in T .
Problem B
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Pi Me m Ct 111 My 6Q{"N~UJ\ node; 0 . Ave
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Homework assignment Number 2
Math 145 / due on 01/19/2007
1. Problems Section 2.5: 1, 5, 7, 8. 2. n persons can arbitrarily shake hands with one another. Show that there is always a pair of persons that shake the same number of hands. 3. Consider a
Homework assignment Number 3
Math 145 / due on 01/26/2007
1. Problems Section 3.8: 6, 8, 9, 10, 13, 15. 2. A "walk" starts at the origin and makes up (resp. down) steps (1, 1) (resp. (1, -1). (a) How many walks are there from the origin to (p, q) for
Homework assignment Number 4
Math 145 / due on 02/02/2007
1. Problems Section 4.3: 5, 6, 8, 9, 11, 13, 15. 2. Let bn be a sequence of numbers such that bo = 0, b1 = 1 and satisfying the recurrence bn+1 = 2bn + 5bn-1 . Find the value of bn . 3. Show
Homework assignment Number 5
Math 145 / due on 02/09/2007
1. Problems Section 7.3: 5, 8, 9, 10, 12. 2. Can we connect 15 computers in such a way that each is connected with exactly 3 other computers? 3. For which values of n does the complete graph K
Homework assignment Number 6
Math 145 / due on 02/16/2007
1. Problems Section 8.5: 2, 3, 4, 5, 6, 7, 8, 9, 12. 2. Consider the graph G = (V, E) where V = {1, 2, 3, 4} and E = {(12), (23), (34), (41), (24)}. Compute the number of spanning trees of G.
Homework assignment Number 8
Math 145 / due on 03/02/2007
1. Exercise 10.1.2. 2. Problems Section 10.4: 5, 7, 10, 11, 13, 15. 3. Problems Section 11.3: 2, 3, 4, 6, 7. Hint for problem 6: show by induction that the maximal number of regions defined b
DEPARTMENT OF MATHEMATICS
SYLLABUS
Course # & Name:
MAT 145 Combinatorics
Recommended Text(s) & Price:
Prepared by:
Kuperberg/Mulase
Lecture(s)
5
4
2
Aprrox. 3 or more
Approx. 2 or more
2
2
Approx. 2 or more
2
Discrete Mathematics, 1st Edition, Lovasz,
Pe