PS05-solutions

# PS05-solutions - MATH 681 Problem Set#5 This problem set is...

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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 ( n 2 n ) in terms of poly- nomials, exponentials, and self-exponentials. You may write it in big-O notation if you wish. By Stirling’s approximation, ± n 2 n ² 1 2 π ( n 2 ) n 2 +1 / 2 ( n 2 - n ) n 2 - n +1 / 2 n n +1 / 2 We can simplify this a great geal with some simple algebra: ± n 2 n ² 1 2 π n 2 n 2 +1 ( n ( n - 1)) n 2 - n +1 / 2 n n +1 / 2 1 2 π n 2 n 2 +1 ( n - 1) n 2 - n +1 / 2 n n 2 +1 1 2 π n ( n 2 ) ( n - 1) n 2 - n +1 / 2 ( n - 1) n - 1 / 2 2 π ± n n - 1 ² ( n 2 ) The ﬁrst term in this product is clearly Θ( n n n ). The second term is less obvious — asymptotically it is a 1 indeterminate form, and experimental veriﬁcation suggests it grows very quickly. On the other hand, we know that ( n 2 n ) < ( n 2 ) n n ! = O ( n 2 n ). Thus, we can, with Stirling’s approximation, get the following (weak) asymptotic bounds: Ω( n n n ) = ± n 2 n ² = O ( n 2 n ) 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? The set S of necklaces prior to symmetry-identiﬁcation is the set of ordered choices of 8 items from a set of size 4, such that each item is chosen at least once. Then | S | is the enumeration statistic 4! S (8 , 4) = ( 4 0 ) 4 8 - ( 4 1 ) 3 8 + ( 4 2 ) 2 8 - ( 4 3 ) 1 8 + ( 4 4 ) 0 8 = 40824. However, we seek | S/D 8 | , where D 8 is the dihedral group consisting of the identity, 7 rotations, and 8 reﬂections. We will invoke Burnside’s lemma to ﬁnd the invariants of each of these permutations. Every bead-coloring would remain ﬁxed under the identity, by deﬁnition, so its invari- ant set is exactly S itself, so it has 4! S (8 , 4) invariants. Page 1 of 5 Due November 10, 2009

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MATH 681 Problem Set #5 Under any of the odd rotations, r , r 3 , r 5 , or r 7 , one may trace the positions of each bead to note that the beads are mapped to eachother in an 8-bead cycle, in which each bead would have to have the same color as its successor, so every bead would have to be the same color; there are no elements of S which match this condition, since every coloring in S uses all 4 colors. Thus r , r 3 , r 5 , and r 7 have no invariants (formally, one might say they have 4! S (1 , 4) invariants, which happens to be zero). Under the rotations
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PS05-solutions - MATH 681 Problem Set#5 This problem set is...

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