Mathematics 501
Final (70 points)
Due 12/14/06, 11.10am
Points (leave blank)
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Name:
(clearly, please)
This exam is my own work. Sources (apart from the textbooks and my lecture notes) are
indicated.
Signature
This final is due Thursday, December 14th 2006 at 11.10am (at my office or in my mailbox).
Notes
• Put your name on this cover sheet and sign it.
• You are permitted to use your notes and any publication (book, journal, web page).
You are not permitted to consult third persons.
Results which are quoted from a publication (apart from the course textbook and your
lecture notes) must be indicated.
• You may use a computer unless this renders the problem trivial.
• If you need extra sheets, staple them to this final. (You do not need to submit scrap
paper.)
• I will put the graded finals with your course grades in your mailboxes in the math
department if you have one. Otherwise you can pick up your exam in the front office
starting the week after finals week.
I can email you final score and grade if you want me to. In this case please write an
email address on the following line. (Due to privacy regulation I cannot honor email
requests for grades if they are not requested here.)
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1) How many ways are there to return 25c of change, using coins in values of 1, 5, 10 and
25c?
(10 points)
2) Let Mλµ be the set of coefficients expressing the elementary symmetric polynomials eλ
in terms of the basic polynomials:
X
eλ =
Mλµ mµ
µ|=n
a) Show that Nλµ is equal to the number of matrices A ∈ {0, 1}n×n which have row sums λ
and column sums µ. (We assume the partitions to be padded with 0 to get n entries. E.g. if
λ = (2, 1) and µ = (1, 1, 1), there are three such matrice, namely:

 
 

1 1 0
1 0 1
0 1 1
 0 0 1 , 0 1 0 , 1 0 0 ;
0 0 0
0 0 0
0 0 0
thus M(2,1),(1,1,1) = 3.
b) Show that Mλµ = Mµλ .
(10 points)
3) Prove that if p is prime we have that
n
bn/pc
n mod p
≡
m
bm/pc m mod p
Hint: Show that (1 + x)pq+r ≡ (1 + x)r (1 + xp )q
(mod p)
(mod p).
(10 points)
4) Let G(t) be the exponential generating function for the recurrence
g0 = 0,
g1 = 1
X n gn = −2ngn−1 +
gk gn−k
k
k
a) Show that G(t) fulfills G(t) = −2tG(t) + G(t)2 + t
b) Solve the recurrence.
(n > 1).
(10 points)
5) Show that there are Cn permutations of the numbers {1, . . . , n} in which the longest
decreasing subsequence has length 2 (i.e. there does not exists i < j < k such that ai > aj >
ak ). For example for n = 3 the permutations are:
123, 231, 132, 312, 231
(Such permutations are called 321-avoiding permutations.)
(10 points)
6) Let n be a positive integer and F a field with q elements. We consider the lattice L
formed by the subspaces of the vector space V := F n ordered by inclusion.
a) Show that for two subspaces U ≤ W ≤ V with l := dim W − dim U there is an inclusionpreserving bijection between the set of subspaces A ≤ V with U ≤ A ≤ W and the set of
subspaces of F l . (Hint: Consider a basis u1 , . . . uk for U and extend it to a basis for W by
adding l further basis vectors w1 , . . . wl .)
b) Show that the Möbius function µ(U, W ) depends only on l := dim W − dim U . We thus
can define µl := µ(0, F l ) and have that µ(U, W ) = µl .
c) Show that µl = −q l−1 µl−1 . (Hint: Show that an l-dimensional vector space has exactly
q l−1 maximal subspaces.)
l
(10 points)
d) Conclude that µl = (−1)l q (2) .
7) Determine the number of permutations g ∈ S9 for which g 2 = 1.
(10 points)
8∗ ) What is the relevance of the following sequence?
un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu, . . .
What would be term 18 in this sequence?
(5 points)