Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 2
1. Case . Fill in the blanks:
# choices :
5
5
5
5
5
4
3
2
5
5
5
2
2
3
4
2
The answer is 54 .
Case cfw_a. Fill in the blanks left to right:
#
MATH413
HW 11 - bonus questions
1:
Does there exist a BIBD with parameters b = 10, v = 8, r = 5, and
k = 4?
2:
Does there exist a BIBD with parameters b = 20, v = 18, r = 10, and
k = 9?
3:
Construct three mutually orthogonal latin squares of order four.
4
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 7
We list some Fibonacci numbers together with their prime factorization.
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
f
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 3
1. For 1 k 22 we show that there exists a succession of consecutive days during which
the grandmaster plays exactly k games. For 1 i 77 let
MATH413
HW 10
due Apr 25 strictly before class
Solutions without explanation will receive no points.
1: P.320, # 22(a) Compute p6 (the partition number of 6) and construct a
Hasse diagram of partialy ordered set P6 where Pn contains all partitions of
n (i
MATH413
HW 6
due Mar 14 before class
1: (P. 199, #13) Determine the number of permutations of cfw_1,2,. . . ,9 in
which at least one odd integer is in its natural position.
Solution:
For i = 1, 3, 5, 7, 9, let Ai denote the set of permutations of cfw_1, .
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 7
We list some Fibonacci numbers together with their prime factorization.
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
f
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 6
1. Dene the set S = cfw_1, 2, . . . , 104 . Let A (resp. B) (resp. C) denote the set of integers in
S that are divisible by 4 (resp. 5) (res
MATH413
HW 1
due Feb 1 before class
1:
How many ways are there to pick a man and a woman who are not
husband and wife from a group of n married couples?
Solution:
Total number of possibilities to pick man and woman is n2 . Number of bad
choices is n. Henc
MATH413
HW 3
due Feb 19 before class, answer without justication will receive 0 points.
1: (P. 66, #50) In how many was can ve identical rooks be placed on
the squares of an 8-by-8 board so that four of them form the corners of a
rectangle with sides para
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 8
1. See the solution to Problem 41 in Chapter 7.
2. Let Pn denote the set of permutations of the multiset cfw_n 1, n 1. Let Mn denote the
set
Math 475
Text: Brualdi, Introductory Combinatorics 5th Ed.
Prof: Paul Terwilliger
Selected solutions for Chapter 5
1. For an integer k and a real number n, we show
n
k
=
n1
n1
+
.
k1
k
First assume k 1. Then each side equals 0. Next assume k = 0. Then eac
Math 413 HW6 Due: Oct 14th
1. (Q30) Find and prove a formula for:
r ,s , t 0
r +s+t =n
( mr )(ms )( mt )
1
2
3
Where the summation extends over all nonnegative integers r, s,
and t with sum r+s+t=n.
The hint suggests that the
r ,s , t 0
r +s+t =n
( mr )(m
U. of Illinois MATH 413 Test 2 Fall 2012
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation will receive no credit. Write your name on the top right
corner of each page.
Notes on partitions and their generating functions
1. Partitions of n.
In these notes we are concerned with partitions of a number n, as opposed to partitions of a set.
A partition of n is a combination (unordered, with repetitions allowed) of positive in
U. of Illinois MATH 413 Test 3 Fall 2012
Answer as many problems as you can. Each of the ve questions is
worth 6 points (The total is out of 30). Show your work. An answer
with no explanation will receive no credit. Write your name on the top
right corner
University of Illinois MATH 413 Test 1 Fall 2009
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right corner of eac
U. of Illinois MATH 413 Practice Test 1 Fall 2009
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right corner of ea
Notes on exponential generating functions and structures.
1. The concept of a structure.
Consider the following counting problems: (1) to nd for each n the number of partitions of an
n-element set, (2) to nd for each n the number of permutations of an n-e
University of Illinois MATH 413 Test 1 (v2) Spring 2010
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right corner
University of Illinois MATH 413 Test 1(v1 and v2) Fall
2012
Answer as many problems as you can. Each question is worth 6 points
(total points is 30). Show your work. An answer with no explanation
will receive no credit. Write your name on the top right co
Exercises on Catalan and Related Numbers
excerpted from Enumerative Combinatorics, vol. 2
published by Cambridge University Press 1999
by Richard P. Stanley
version of 23 June 1998
,
1
19. 1 3+ Show that the Catalan numbers Cn = n+1 2nn count the number
MATH413
HW 6
due Mar 14 before class
1: (P. 199, #13) Determine the number of permutations of cfw_1,2,. . . ,9 in
which at least one odd integer is in its natural position.
2: (P. 199, #17) Determine the number of permutations of the multiset
S = cfw_3 a,
MATH413
HW 4
due Feb 29 before class
1: (P. 155, #8) Use binomial theorem to prove that
n
X
n nk
(1)k
2n =
3
k
k=0
2: (P. 155, #11) Use combinatorial reasoning to prove the identity (in
the given form)
n
n3
n1
n2
n3
=
+
+
k
k
k1
k1
k1
3: (P.155
MATH413
HW 1
due Feb 1 before class
1:
How many ways are there to pick a man and a woman who are not
husband and wife from a group of n married couples?
2: How many nonempty words can be formed from three As and five Bs?
(not all letters must be used)
3: