Question 1 (Permutation Notation, 30 points). Consider the string S = 425613.
(a) If S is interpreted as a permutation of [6] in the standard notation, how would you write this permutation
in canonical cycle form? [15 points]
When considered as a permutat
Math 184A Homework 3
Fall 2016
This homework is due on gradescope by Friday October 28th at 11:59pm. Remember to justify your
work even if the problem does not explicitly say so. Writing your solutions in LATEXis recommend though
not required.
Question 1
Math 184A HW 4 Solutions
Fall 2015
Question 1 (Permutation Parity, 40 points). .
(a) Show that for any n > 1 that the number of permutations of [n] with an even number of cycles is equal
to the number of permutations of [n] with an odd number of cycles us
Math 184A Homework 2
Fall 2016
This homework is due on gradescope by Friday October 14th at 11:59pm. Remember to justify your
work even if the problem does not explicitly say so. Writing your solutions in LATEXis recommend though
not required.
Question 1
Question 1 (Eulerian Trail, 15 points). Either exhibit an Eulerian trail (not necessarily closed) in the graph
below or show that one does not exist:
B
H
C
D
F
A
E
G
We note that ABGEADEF HECBHC is such a path.
1
Question 2 (Restricted Permutation Countin
Math 184A Homework 3
Fall 2016
This homework is due on gradescope by Friday November 11th at 11:59pm. Remember to justify your
work even if the problem does not explicitly say so. Writing your solutions in LATEXis recommend though
not required.
Question 1
Math 184A, Winter 2017, Prof. Tesler
Homework #4, Due Wednesday February 8, 2017
Chapter 5# 32 (B(n) is the Bell number, defined on page 97; in class, we used notation Bn .)
Chapter 8# 2 , 7, 23, 35
and the problems below, H-9 through H-12 (see both pages
Math 184A, Winter 2017, Prof. Tesler
Homework #3, Due Wednesday February 1, 2017
and the problems below, H-3 through H-8 (see both pages).
Chapter 5# 6, 8, 12 , 26, 27;
Note: On Chapter 5# 12, your answer should be in terms of the function p(n); you dont
Math 184A, Winter 2017, Prof. Tesler
Homework #5, Due Wednesday February 15, 2017
Note: If you would like to use your homework to study for the exam, please photocopy
it before turning it in.
Chapter 7# 3, 4, 7, 19, 24, 25, 26, 28
Chapter 8# 12, 39
and th
Math 184A Homework 5 Solution
Question 1 (Combinatorial Identity, 20 points). Come up with a combinatorial proof of the following identity
for n 2m > 0:
nm
X n
2m
c(k, m)c(n k, m) =
c(n, 2m).
k
m
k=m
Solution. We count the same number twice to prove thi
Homework 1 Solutions
Question 1 (Fibonacci Numbers, 30 points). The Fibonacci Numbers is the sequence 1, 1, 2, 3, 5, 8,
13 . . . defined as follows:
F1 = F2 = 1
F n+2 = F n+1 + F n for all n 1
(a) Prove that F n+2 counts the number of strings of length n
Math 184A Homework 2 Solution
Question 1 (Colored Balls in Bins, 30 points). Consider the problem of placing unlabelled red and blue
colored balls into labelled bins. As the balls are unlabelled you cannot distinguish one from the other except
by color. Y
Homework 4 Solutions
Question 1 (Stirling Number Computation). Compute the values of c(n, k) for all 1 k n 6.
Answer: c(n, n) = 1, since the only way to split [n] into n cycles is to have each cycle a
singleton, which gives the identity permutation. From
MATH 184A
HW3 Solutions
1.a. By relabeling we can assume that a1 = n. We then consider a permutation in
which a1 , . . . , ak are in the same cycle. When we put this permutation into canonical
cycle notation, the cycle containing a1 , . . . , ak will star
Math 184A Homework 3
Fall 2015
This homework is due Monday October 26th in discussion section. Remember to justify your work even
if the problem does not explicitly say so. Writing your solutions in LATEXis recommend though not required.
Question 1 (Diffe
Solution for Homework 6
Fall 2015
This homework is due Monday November 23rd in discussion section. Remember to justify your work
even if the problem does not explicitly say so. Writing your solutions in LATEXis recommend though not
required. If you cannot
Math 184A Homework 1
Fall 2015
This homework is due Monday October 5th in discussion section. Remember to justify your work even if
the problem does not explicitly say so. Writing your solutions in LATEXis recommend though not required.
Question 1 (Summat
Math 184A Final Exam
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 180 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. You are free to make use
of any result in t
Math 184A Exam 2
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 45 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. In particular, in order to
get full credit, you
Math 184A Homework 4
Fall 2015
This homework is due Monday November 2nd in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Optional Practi
Math 184A Homework 5
Fall 2015
This homework is due Monday November 9th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Optional Practi
Math 184A Homework 6
Fall 2015
This homework is due Monday November 23rd in discussion section. Remember to justify your work
A
even if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not
required. If you cannot
Math 184A Homework 3
Fall 2015
This homework is due Monday October 26th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Die
Math 184A Exam 1
Fall 2015
Instructions: Do not open until the exam starts. The exam will run for 45 minutes. The problems are
roughly sorted in increasing order of diculty. Answer all questions completely. In particular, in order to
get full credit, you
Math 184A Homework 1
Fall 2015
This homework is due Monday October 5th in discussion section. Remember to justify your work even if
A
the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Summ
Math 184A Homework 2
Fall 2015
This homework is due Monday October 12th in discussion section. Remember to justify your work even
A
if the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not required.
Question 1 (Equ
Math 184A HW 2 Solution
Question 1 (Equal Sum Subsets, 20 points). Let S be a set of 10 positive integers each at most 100. Show
that there exist two dierent subsets A S and B S so that the sum of the elements of A equals the sum
of the elements of B.
Ans