Download MATH 4707 PROBLEM SET 2 1. Required problems

MATH 4707 PROBLEM SET 2
FINAL VERSION; DUE WEDNESDAY 2/18
1. Required problems
• Exercises from the text: 2.5.8, 3.2.2(a–c), 3.2.3 (in part (a), “n” should be 8), 3.8.6,
3.8.12, 3.8.13,
(1) (a) How many (n + 1)-element subsets are there of the set {1, 2, . . . , n + k + 1}?
(b) Let i be some integer between n + 1 and n + k + 1. Of the subsets in the previous
part, how many of them have largest element i?
(c) Write down an equation relating the answers of the previous two parts. (It should
probably involve a sum.)
(d) What does this have to do with Figure 3.3 (page 53) in the text?
(2) Let Mn be the (n + 1) × (n + 1) matrix with rows and columns numbered 0, 1, . . . , n
such that the entry in row i and column j is ji . Compute the inverse matrices of M0 ,
M1 , M2 and M3 . (Feel free to use software for this step.) What do you notice? (On this
exercise only, no explanation required. But, you might reflect on the relationship with
Inclusion-Exclusion.)
(3) (10 points) Let U denote the set of all permutations of the set [n] = {1, 2, . . . , n}. For
1 ≤ i ≤ n − 1, let Ai denote the subset of U consisting of those permutations in which i
is followed immediately by i + 1.
For example, when n = 3, U is the set containing the six permutations 123, 132, 213,
231, 312 and 321. The set A1 consists of those permutations in which 1 is immediately
followed by 2, namely 123 and 312. The set A2 consists of those permutations in which 2
is immediately followed by 3, namely 123 and 231. Their intersection is A1 ∩ A2 = {123},
while the three permutations 321, 132 and 213 belong to neither A1 nor A2 .
(a) In the case n = 4, explicitly list the 6 elements of A1 and indicate which belong to
A1 ∩ A2 and which belong to A1 ∩ A3 . (It may be helpful to think of the condition
on the elements of A1 as “gluing” the 2 to the 1 in order to form a single block
“12”.)
(b) Now consider n an arbitrary integer larger than 3. What are |A1 |, |A2 |, |A1 ∩ A2 |,
and |A1 ∩ A3 |? (Of course, your answer
T will have an n in it!)
(c) In general, for I ⊆ [n − 1], T
what is i∈I Ai ? (Recall that if I = {i1 , i2 , . . . , ik } for
integers k, i1 , . . . , ik , then i∈I Ai = Ai1 ∩ Ai2 ∩ · · · ∩ Aik is the set of elements that
belong to all of Ai1 , Ai2 , . . . . Again, it may be helpful to think of Aij as being the
set of permutations in which the element (ij + 1) is “glued” immediately after the
element ij as part of the same block.) [Hint: the answer will depend only on the
size |I| of the set I, not on the precise values of the elements of I.]
(d) Write a sentence in English describing the set (A1 ∪ A2 ∪ · · · ∪ An−1 )C .
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FINAL VERSION; DUE WEDNESDAY 2/18
(e) Compute the size (A1 ∪ A2 ∪ · · · ∪ An−1 )C of this set. (Try to express the answer
in some not-too-ugly form; say, something simpler than a sum over all subsets of a
set.)
(f) Compare your answer in part (e) for n = 3 to the second paragraph of the problem
statement.
(4) Exercise 4 from Dennis White’s handout
• Exercises from the text: 2.5.5, 4.1.3, 4.2.4
2. Optional problems
• Draw the first 4, 8 or 16 rows of Pascal’s triangle. Mark those entries that are even in
one color and those that are odd in another color. What pattern do you observe? (Note:
since knowing the parity of two numbers is enough information to know the parity of
their sum, you can do all of the arithmetic modulo 2, just keeping track of the parity of
the entry in each row to get the parity of the entries in the next row.)
• Exercises from the text: 2.5.7, 3.6.3, 3.7.1, 3.8.8, 3.8.9, 3.8.11, 3.8.14, 4.2.5, 4.3.4, 4.3.7
• Exercises 1, 2, 3 and 5 from Dennis White’s handout
• Suppose that I tell you that a sequence (an ) satisfies a0 = 0 and a9 = 34 and an+1 =
an + an−1 for n ≥ 1. Does this uniquely define the sequence?
• Putnam 1985 A1: Determine, with proof, the number of ordered triples (A1 , A2 , A3 ) of
sets which have the property that A1 ∪ A2 ∪ A3 = {1, 2, . . . , 10} and A1 ∩ A2 ∩ A3 = ∅.
• Putnam 1993 A3: Let Pn be the set of subsets of {1, 2, . . . , n}. Let c(n, m) be the number
of functions f : Pn → {1, 2, . . . , m} such that f (A ∩ B) = min(f (A), f (B)). Prove that
m
X
c(n, m) =
j n.
j=1
• Putnam 1990 A6: Call an ordered pair (S, T ) of subsets of {1, 2, . . . , n} admissible if
s > |T | for each s ∈ S and t > |S| for each t ∈ T . How many admissible ordered pairs
of subsets of {1, 2, . . . , 10} are there?