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 =
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 th
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 equall
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
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 Assignmen
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.
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 p
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 t
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 i
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
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
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
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 als
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 re
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
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 Assi
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 p
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
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
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 ,
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