Math 462
Solutions for Homework 4
February 4, 2013
1
1. (This is #27 on p. 126) Find the number of permutations of six elements whose square
is the identity.
Solution: For a permutation to satisfy 2 = id, each cycle of must be of length 1
or 2. Thus all s
Math 462
Solutions for Homework 7
March 6, 2017
1
1. Determine the number of regions that are created by n lines in the plane if it is known that (i)
no two of these lines are parallel, and (ii) no three lines share a common point.
Solution: Denote this n
Math 462
Solutions for Homework 6
February 24, 2017
1
1. Let be the set of all strings of length n in the alphabet cfw_1, 2, 3. We make into a probability
space by assuming that all strings are equally likely.
(a) What is the expected number of 1s in a ra
Math 462
Solutions for Homework 4
February 3, 2017
1
1. (a) Let cfw_A1 , . . . , As be a collection of l-subsets of [n], where l < n/2. Show that there exist
s distinct (l + 1)-subsets B1 , . . . Bs of [n] such that Ai Bi for every 1 i s.
Solution: As in
Math 462
Homework #7
February 24, 2017
1
This week well discuss recurrence relations and generating functions. Please read the lecture notes and
take a look at 12.1-12.3 in the book.
Written Assignments Due Monday, 3/6/17.
1. Determine the number of regio
Math 462
Homework #3
January 20, 2017
1
This week we will discuss the Sperner theorem (see 7.2) and the Erdos-Ko-Rado theorem. Please read
the lecture notes as well as take a look at 7.2 in the book.
Written Assignments Due Friday, 1/27/17.
1. Let k, l be
Math 462
Solutions for Homework 5
February 17, 2017
1
1. Assume you have a 6-sided dice with numbers from 1 to 6 written on its sides. Also assume that
the dice is unbalanced: when you throw it, the probability to get a number k is proportional to
k. What
Math 462
Solutions for Homework #1
January 13, 2017
1
1. Is there a bipartite graph with degree sequence (3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 6, 6, 6)?
Solution: Note that in the given degree sequence all the numbers are divisible by 3 except
one. Hence if we p
Math 462A
Solutions for Review problems
March 11
1
1. An n n matrix is called a permutation matrix if all entries are 0 or 1, and there is
precisely one 1 in each row and each column. Show that if M is an n n matrix with all
entries 0 or 1, and with exact
FINAL EXAM
Math 462
Your Signatnre
Ytxrr Ntune
Student
ID
March 75, 2017
f
Problem
Points
Possible
Problem
1
8
4
b
2
b
5
6
3
t4
6
8
Total
,18
o No books allowed. You may
o Do not share
o Make
use one
Points
8| x 11 sheet (double-sided)
Pcissible
of notes.
Math 462
Solutions for Homework 3
January 27, 2017
1
1. Let k, l be natural numbers. Prove that every sequence of real numbers of length kl + 1
contains a nondecreasing subsequence of length k + 1 or a decreasing subsequence of length
l + 1.
Solution: We
Math 462
Homework #1
January 4, 2017
1
This week we will discuss bipartite graphs and matchings in bipartite graphs. The most
important theorem well learn is Philip Halls theorem (also known as Halls marriage theorem).
Please read all the lecture notes; s
Math 462
Solutions for Homework #2
January 20, 2017
1
1. (a) Let 1 and 2 be two partial orders on a set X. Define a new relation on X by x y if and
only if both x 1 y and x 2 y hold. Prove that is also a partial order.
Solution: We are given that the rela
Math 462
Homework #6
February 25, 2013
1
This week well discuss Chapter 15: please read 15.1, 15.2, 15.4; you may skip 15.4.3 for
now.
Written Assignments Due Monday, 3/4/13.
1. A deck of cards has regular 52 cards and 2 jokers (54 cards total). Assume th
Math 462
Solutions for Homework 3
January 28, 2013
1
1. (This is #33 on p. 107) Let F (n) be the number of all partitions of [n] with no singleton
blocks. Find the recursive formula for the numbers F (n) in terms of the numbers F (i),
with i n 1.
Solution
Math 462
Homework #4
January 28, 2013
1
This week well talk a bit more about Stirling numbers of the 1st kind (see 6.1). Well
then start discussing Chapter 11 (please read 11.1 and 11.2).
Written Assignments Due Monday, 2/4/13.
1. (This is #27 on p. 126)
Math 462
Solutions for Homework 5
February 25, 2013
1
1. Let G be a bipartite graph all of whose vertices have the same degree d. Show that there
are at least d distinct perfect matchings in G. (Two perfect matchings M1 and M2 are
distinct if M1 = M2 as s
Math 462
Homework #3
January 23, 2013
1
This week well talk about permutations and Stirling numbers of the 1st kind (see 6.1).
Written Assignments Due Monday, 1/28/13.
1. (This is #33 on p. 107) Let F (n) be the number of all partitions of [n] with no
sin
Math 462
Solutions for Homework 1
January 14, 2013
1
1. Determine the number of regions that are created by n lines in the plane if it is known that
(i) no two of these lines are parallel; and (ii) no three lines share a common point.
Solution: Denote thi
Math 462
Solutions for Homework 2
1
January 23, 2013
1. Determine the generating function hn for the number of nonnegative integral solutions
of 2e1 + 5e2 + e3 + 7e4 = n.
Solution: By denition of hn , we have
x2e1 +5e2 +e3 +7e4 =
hn xn =
n=0
e1 =0
(e1 ,e2
Math 462
Homework #2
January 14, 2013
1
This week we will study products of generating functions (8.1.2) and Catalan numbers
(8.1.2.1); after that well review Stirling numbers of the 2nd kind and will discuss Bell
numbers (5.2). Next week well talk about
Math 462
Homework #2
January 13, 2017
1
This week we will discuss partially ordered sets. Please read the lecture notes as well as
Chapter 2 in the book.
Written Assignments Due Friday, 1/20/17.
1. (a) Let 1 and 2 be two partial orders on a set X. Define
Math 462
Homework #6
February 17, 2017
1
We continue discussing the probabilistic method. Please read the lecture notes and 10.2 (but skip the
subsection Independent events) and 10.3, 10.4 in the book.
Written Assignments Due Friday, 2/24/17.
1. Let be th
Math 462
Homework #5
February 13, 2017
1
Our next few lectures will be devoted to the probabilistic method. Specifically, well discuss random
variables and expected values, as well as applications of expected values to combinatorics. Please read
the lectu
Math 462
Course Information
January 4, 2017
1
Instructor: Prof. Isabella Novik, e-mail [email protected]
Office: Padelford C-416, (206)616-9373
Class web-page: http:/www.math.washington.edu/novik/462/
Lectures: CDH 125, MWF 11:3012:20
Teaching Ass
Math 462
Homework #4
January 27, 2017
1
This week we will dicsuss Ramseys theorem. Please read the lecture notes as well as 11.1 and 11.2
in the book.
Written Assignments Due Wednesday, 2/3/17.
1. (a) Let cfw_A1 , . . . , As be a collection of l-subsets