Math 154
Assignment 2
[email protected]
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
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 (
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 .
The
Principle of
Inclusion

Exclusion
.
Fundamental Problem
Combinator
( finite )
151
5
in
ics
:
given
a
set
S
,
determine
.
.
Different strategies
If
S
strategy
is
,
contained
depending
in
a
on
how
Test 1, Math 154 Discrete Mathematics and Graph Theory
Problem 1 (20 points). Prove (from scratch) the ErdosSzekeres monotone subsequence
theorem: Every sequence a1 , a2 , . . . , an2 +1 of n2 + 1 di
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
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 nd
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 ver
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
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=
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. Det
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 witho
Homework 2 Solutions, Math 154
3.4. Partition cfw_1, 2, ., 2n into n parts cfw_1, 2, cfw_3, 4, ., cfw_2n 1, 2n. By pigeonhole, any
n + 1 numbers selected must have two from the same part chosen. These
Practice Test 2, Math 154
Problem 1 Does there exist a (simple) graph of order 5 whose degree sequence is (4, 4, 4, 2, 2)?
Is there a general graph with no loops whose degree sequence is (4, 4, 4, 2,
Homework 1 Solutions, Math 154
2.1. With no restrictions, there are 54 solutions. For (a) to hold we 5432 = 120 solutions.
For (b) to hold we have 2 53 = 250 since the last digit must be a 2 or 4. For
Math 154
Solutions to Assignment 1
[email protected]
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,
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
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. Fi
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 u
Math 154: Winter 2017
Homework 1
Due 5:00pm on Wednesday 1/18/2017
Problem 1: How many ways are there to place n nonattacking rooks on an n n
chessboard?
Problem 2: For any set A, let Ak be the set o
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 nonattacking rooks on an n n
chessboard?
Solution: We claim that there are n
Rhoades Math 154: Winter 2017
Homework 6
Due 5:00pm on Wednesday 3/1/2017
Problem 1: Let W be a simple graph with n+1 vertices, each of which has degree d.
Prove that
n
(W)
.
nd
Problem 2: Let G be
Math 154: Winter 2017 Rhoades
Homework 7
Due 5:00pm on Wednesday 3/15/2017
Pr ob lem 1: Show, b y finding an or ient ation on e ach, that the c omplete
n graph A
(n 3) and the c omplete b ipartite gra
Math 154: Winter 2017 Rhoades
Homework 5
Due 5:00pm on Wednesday 2/22/2017
Problem 1: Let W be a polyhedron (or polyhedral graph), each of whose faces is
bounded by a pentagon or a hexagon. Use Eulers
Practice Test 1
MCS 421 Combinatorics
Problem 1. How many sets of three integers between 1 and 20 are possible if no two consecutive
integers are to be in a set?
Problem 2. A ferris wheel has five car