MATH 681 Problem Set #5
This problem set is due at the beginning of class on November 10.
1. (5 points) Find an asymptotically accurate approximation for n2
nin terms of poly-
nomials, exponentials, and self-exponentials. You may write it in big-O notation if you
wish.
2. (10 points) You have a large supply of beads of 4 different colors and want to string
eight of them on a necklace, making use of each bead at least once. How many ways are
there to do so, if necklaces are considered identical if they are rotations or reflections
of each other?
3. (25 points) Answer the following questions about icosohedron-coloring. You may find
it useful to assemble the model at
http://www.korthalsaltes.com/model.php?name_en=icosahedron in order to help
your visualization.
(a) (5 points) Identify the 60 rotation-permutations of the icosohedron. You need
not explicitly give all 60; merely give a classification scheme which identifies 60
different rotations.
(b) (10 points) Using your above rotations and Burnside’s lemma, determine how
many distinct ways there are to color the 20 faces of the icosohedron with 2 colors
if two colorings are regarded as identical if they are rotations of each other. Note:
this calculation will include large exponents; you may use a computer to calculate
or leave them unreduced. Then, generalize your result to indicate how many ways
there are to color the faces of an icosohedron with ncolors.
(c) (10 points) How many ways are there to color the vertices of an icosohedron
with 2 colors? With ncolors?
4. (5 point bonus) Let Abe a set of n-colorings of a p-gon, where pis prime, such that
if a coloring Xis in A, so is every rotation of X. Recalling that A/Cnis the set of
equivalence classes of Aunder rotation, prove that |A|
p≤ |A/Cn| ≤ |A|+(p−1)n
p.
I was seriously tormented by the thought of the exhaustibility of musical combinations.
The octave consists only of five tones and two semi-tones, which can be put together
in only a limited number of ways, of which but a small proportion are beautiful: most
of these, it seemed to me, must have been already discovered, and there could not be
room for a long succession of Mozarts and Webers, to strike out, as these had done,
entirely new and surpassingly rich veins of musical beauty. —John Stuart Mill
Page 1 of 1 Due November 10, 2009