Math 154
Assignment 2
jacques@ucsd.edu
This assignment carries a total of twenty points, where each question has equal weight. The
assignment is due on Friday April 24th by noon.
Question 1. Determine each of the following quantities explicitly.
(a)
1
2
4
3
Recurrence Equations
The generating function for the set of binary strings with no block of ones of odd length
was shown to be
1
(x) =
1 x x2
in the last section. We could nd the coecient of xn in (x) using the binomial theorem,
and the answer would be
MATH 154 Homework 5 - Solutions
Due December 7, 2012
Version December 10, 2012
Assigned questions to hand in:
(1) Suppose a drawer contains ten red beads, eight blue beads, and eleven green beads. Determine a generating function that encodes the answer to
Practice Midterm 1 Math 154
Time: 40 Minutes | No notes allowed
Questions carry equal weight | Calculators allowed
Question 1. State the Binomial Theorem, and use it to prove
n
k
2 n
(1 + x + x ) =
k=0 j=0
1
n
k
k j+k
x .
j
Question 2. Prove by induction
Solutions to Practice Midterm Math 154
Time: 40 Minutes | No notes allowed
Questions carry equal weight | Calculators allowed
The grade is based on your three best questions
Question 1.
A walk on the integers starts at zero and then at each step moves one
Solutions to Practice Final 1
Math 154 Combinatorics and Graph Theory
Instructor J. Verstraete
Allotted time 3 hours
Answers are to be written clearly and legibly
State clearly any theorems used without proof
Calculators are permitted
Total 50 points
Ques
5
Flows and cuts in digraphs
Recall that a digraph or network is a pair G = (V, E) where V is a set and E is a multiset
of ordered pairs of elements of V , which we refer to as arcs. Note that two vertices can
be joined by many arcs in either direction. I
2
Generating Functions
In this part of the course, were going to introduce algebraic methods for counting and
proving combinatorial identities. This is often greatly advantageous over the method
of nding bijections, where it may not even be clear what eac
1
Basic Combinatorics
1.1
Sets and sequences
Sets. A set is an unordered collection of distinct objects. The objects are called elements
of the set. We use braces to denote a set, for example, the set with elements 1, 2 and 3 is
denoted cfw_1, 2, 3. Since
1.9
Exercises
Question 1. Determine the number of sequences (x1 , x2 , . . . , xk ) where xi [n] with the
given restrictions.
(a)
(b)
(c)
(d)
For
For
For
For
i n,
i n,
i n,
i n,
xi is odd.
xi = xi+1 .
xi+1 > xi .
xi+1 > xi + 1.
Figure 4 : Square and trian
Math 154: Winter 2017
Homework 1 Solutions
Due 5:00pm on Wednesday 1/18/2017
Problem 1: How many ways are there to place n non-attacking rooks on an n n
chessboard?
Solution: We claim that there are n! such rook placements. To see this, number
the rows of
Math 154: Winter 2017
Homework 1
Due 5:00pm on Wednesday 1/18/2017
Problem 1: How many ways are there to place n non-attacking rooks on an n n
chessboard?
Problem 2: For any set A, let Ak be the set of k-element subsets of A:
A
:= cfw_S A : |S| = k.
k
W
2.14
Exercises
Question 1. Determine the exact numerical values of the following binomial coecients:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
1
2
2
0
n
n1
1/2
4
1/2
4
1/3
3
1/3
k .
Question 2. Find the inverses of the following formal power series as a sum of powers o
Math 154
Solutions to Assignment 1
jacques@ucsd.edu
Question 1. Determine the number of sequences (x1 , x2 , . . . , xk ) where xi [n] with the
given restrictions.
(a)
(b)
(c)
(d)
For
For
For
For
i n,
i n,
i n,
i n,
xi is odd.
xi = xi+1 .
xi+1 > xi .
xi+1
A more general fact is left as an exercise: if A is any doubly stochastic matrix, and
the rows and columns sum to 1, then A is contained in the convex hull of the set of
permutation matrices, namely
A = 1 P1 + . . . + N PN
where Pi is a permutation matrix
Math 154
Solutions to Assignment 4
jacques@ucsd.edu
Question 1.
(a) (G) = (G) = (G) = 2, (u, v) = 2 = (u, v).
(b) (G) = 5, (G) = 2, (G) = 4, (u, v) = 2, (u, v) = 4.
Question 4. If the graph is complete, then it is Kn for some n 4, which clearly contains
a
Math 154: Winter 2017
Homework 2
Due 5:00pm on Wednesday 1/25/2017
Problem 1: How many subsets of cfw_1, 2, . . . , n do not contain any consecutive numbers?
Problem 2: Let n be a positive integer. Find a proof of the identity
n
X
n
k(k 1)
= n(n 1)2n2
k
Math 154: Winter 2017
Homework 3
Due 5:00pm on Wednesday 2/1/2017
Problem 1: Let G be a simple graph. The complement G of G has the same vertex
/ E(G). A graph is called
set as G, and edges given by uv E(G) if and only if uv
self-complementary if it is i