Math 249
Assignment 1
Due: Wednesday, September 18
1. (5 points) Use induction on n and the recurrence
n
n 1
n 1
=
+
k
k
k 1
to prove the binomial theorem:
(1 + x )n =
n
k =0
Solution: Assume that
(1 + x )n 1 =
n
k =0
nk
x.
k
n 1 k
x.
k
Then
n
(1 + x )n =
MATH 249 Winter 2015 Homework #5
due Friday, March 27th, 2015.
Short notice, so a short problem set. (I left o some good questions,
and the pre-pre-lemma: if there are two distinct paths from v to w
then G contains a cycle.)
1. Give an example of a plane
MATH 249 Winter 2015 Homework #3
Due at the start of class, Friday Feb. 13th.
Questions 1 and 2 are human-sized questions that can be done by
hand without grief. For question 2, you can use some computer algebra
software (Maple, Matlab, Mathematica,.) if
MATH 249 Winter 2015 Homework #2
Due at the start of class, Friday Jan. 30th.
1. Fix a positive integer p 1. For n 1, among all 2n1 compositions
of size n, what is the average number of parts of size p?
2(a) Let An be the set of all compositions of size n
MATH 249 Winter 2015 Outline
MATH 249 has two parts: rst will be Enumeration and second
will be Graph Theory. We will cover everything in MATH 239, and
plenty of extras.
Enumeration.
Rational power series, recurrence relations, partial fractions. Binomial
MATH 249 Winter 2015 Homework #6
due Monday, April 6th, 2015.
April 3rd is Good Friday, a University holiday. April 6th is the last
day of classes.
1. The cube is a 3-regular bipartite plane graph with p = 8 vertices.
Draw an example of a 3-regular bipart
Figure 1. Two drawings of the Petersen graph.
Figure 2. Two are the same, one is dierent.
MATH 249 Winter 2015 Homework #4
due Monday, March 9th.
REMINDER: the midterm exam is on Thursday, March 5, 4:40-6:20.
I will tell you the room number when I nd out.
MATH 249 Winter 2015 Midterm Solutions.
There are ve questions for a total of 35 marks.
question 1. 2. 3. 4. 5.
marks 6 6 6 9 8
1.
Consider the power series
1
=
1 x(1 + xy)
pn (y)xn
n=0
in which each pn (y) is a polynomial in y.
(a) [5pts] Obtain a formul
MATH 249 Winter 2015 Homework #1 Solutions
1. Dene a sequence of numbers cfw_cn : n 0 by the initial conditions c0 =
1, c1 = 2, and c2 = 3, and the recurrence relation cn = cn1 +2cn2 +2cn3
for all n 3. Obtain an algebraic formula for the rational function
Basic Principles of Enumeration.
In this section and the next well see several basic principles of enumeration
and apply them to a variety of problems, of both mathematical and general interest.
The basic principles themselves are phrased in the language
MATH 249 Winter 2015 Homework Solutions #2
1. Fix a positive integer p 1. For n 1, among all 2n1 compositions
of size n, what is the average number of parts of size p?
A single part in such a composition is a positive integer c P = cfw_1, 2, 3, ..
The gen
MATH 249 Winter 2015 Homework #3
Due at the start of class, Friday Feb. 13th.
Questions 1 and 2 are human-sized questions that can be done by
hand without grief. For question 2, you can use some computer algebra
software (Maple, Matlab, Mathematica,.) if
Figure 1. Two drawings of the Petersen graph.
Figure 2. The spokes of the Petersen graph.
MATH 249 Winter 2015 Homework #4 Solutions
1.
Show that the Petersen graph does not have a Hamilton cycle.
Consider the picture of the Petersen graph (Pete) in Figur
Figure 1. Question 1 not quite.
MATH 249 Winter 2015 Homework #6
due Monday, April 6th, 2015.
April 3rd is Good Friday, a University holiday. April 6th is the last
day of classes.
1. The cube is a 3-regular bipartite plane graph with p = 8 vertices.
Draw
Partial Fractions Expansion
Theorem: Let F (x) = P (x)/Q(x) be a quotient of polynomials in which deg P <
deg Q and the constant term of Q(x) is 1. Factor the denominator to obtain its
inverse roots:
Q(x) = (1 1 x)d1 (1 2 x)d2 (1 s x)ds
where 1 , ., s are
Figure 1. Example for Question 1.
MATH 249 Winter 2015 Solutions #5
due Friday, March 27th, 2015.
Short notice, so a short problem set. (I left o some good questions,
and the pre-pre-lemma: if there are two distinct paths from v to w
then G contains a cyc
14. Enumeration and Symmetry.
Another type of enumeration problem to which neither ordinary nor exponential generating functions apply directly involves enumerating symmetry classes
of combinatorial objects. (In fact, some of the theory of exponential gen
MATH 249 Winter 2015 Homework #1
Due at the start of class, Friday Jan. 16th.
1. Dene a sequence of numbers cfw_cn : n 0 by the initial conditions c0 =
1, c1 = 2, and c2 = 3, and the recurrence relation cn = cn1 +2cn2 +2cn3
for all n 3. Obtain an algebrai
Math 249, Winter 2013
Assignment 6
Due Wednesday, March 20, in class.
1. Let p N, and let n1 , n2 , n3 , . . . , np be a sequence of non-negative integers such that
p
p
ni = p
i=1
ini = 2p 2 .
and
i=1
Prove that there exists a tree with p vertices that ha
Math 249
Assignment 2
Due: Wednesday, September 25
1. (5 points) If F (x ) = f n x n , prove the the coefcient of x n in (1 x )1 F (x ) is n=0 f i . Using
i
this, and the binomial formula applied to (1 x )2 and (1 x )3 , show that the sum of the
rst n pos
Math 249
Assignment 3
Due: Wednesday, October 2
1. (7 points) A sequence (c n ) of integers is given by the initial conditions c 0 = c 1 = 1 and the
recurrence
c n = c n 1 + 2c n 2
when n 2. Show that the generating series for c n is (1 x 2x 2 )1 and usin
Math 249
Assignment 4
Due: Wednesday, October 9
1. (7 points) In the string game from the lectures and notes, determine the probability that
the string aba occurs before bba .
Solution: Let L denote the set of strings that contain no copies of aba or bba
Math 249
Assignment 5
Due: Wednesday, October 16
1. (5 points) Prove that the number of self-conjugate partitions of n is equal to the number of
partitions of n with distinct odd parts, and hence write down their generating series.
Solution: The generatin
Math 249
Assignment 6
Due: Wednesday, October 23
1. (5 points) If A and B are q -commuting variables, show that expq ( A + B ) = expq ( A ) expq (B ).
2. (5 points) [withdrawn]
3. (5 points) Dene
P (t , x ) =
1
n
n 1 1 t x
Let p e (n ) and p o (n ) denote
Math 249
Assignment 6
Due: Wednesday, October 23
1. Let ( f n )n 0 and (g n )n 0 be sequences with generating series F (x ) and G (x ) respectively and
suppose that there are constants a 1 , . . . , a k such that
f n + a 1 f n 1 + + a k f n k = g n ,
n k.
MATH 249 NOTES
Ian Goulden April 4, 2008
1
Lecture of January 9
The rst six weeks of the course will be concerned with Enumerative Combinatorics, also referred to as Enumeration, Combinatorial Analysis or, simply, Counting. This subject concerns the basic
MATH 239: Introduction to Combinatorics
Chris Thomson
Winter 2013, University of Waterloo
Notes written from Bertrand Guenins lectures.
See
cthomson.ca/notes. Last modied: August 14, 2013 at 11:15 PM (f33d815).
1
Contents
1 Introduction, Permutations, an
Math 249, Winter 2013
Assignment 1
Due Wednesday, January 23, in class.
1. Let X be a nite, non-empty set. Prove that the number of odd subsets X is equal to the
number of even subsets of X .
(An odd subset of X is a subset with an odd number of elements.
Math 249, Winter 2013
Assignment 3
Due Wednesday, February 13, in class.
1. Let S be a set of combinatorial objects with a weight function. Prove that the number of
elements of S that have weight at most n is [xn ]S (x)(1 x)1 .
2. How many ways are there