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4203hw4 - sidered diFerent 4 Let G n be the number of(set...

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Combinatorics (MAD 4203) — Fall 2010 Due Wednesday 10/20 Homework #4 1 Without using any asymptotic bounds for p ( n ), prove that p ( n ) grows faster than any polynomial. That is, if f ( n ) is any polynomial, prove that there is some integer N such that p ( n ) > f ( n ) for all n > N . You may submit an answer to this problem as long as you did not receive full credit for it on the last homework. 2 How many permutations π of [6] satisfy π 3 = 1? 3 A group of 10 children want to play cards. They split into three groups, one with 4 children and the other two with 3 children. Then each group sits around a table. Two seatings are considered the same if everyone’s left neighbor is the same. a. In how many ways can this be done if the three tables are con- sidered identical? b. In how many ways can this be done if the three tables are con-
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Unformatted text preview: sidered diFerent? 4 Let G ( n ) be the number of (set) partitions of [ n ] without singleton blocks. Prove that B ( n ) = G ( n ) + G ( n + 1). 5 De±ne the family { P n ( x ) } of polynomials by P n ( x ) = n s k =0 S ( n,k ) x k , where S ( n,k ) denotes a Stirling number of the second kind. Prove that P n ( x ) = x p d dx P n-1 ( x ) + P n-1 ( x ) P . 6 Let π be a permutation of [ n ]. An inversion of π is a pair of indices i < j such that π ( i ) > π ( j ). A permutation is even if it has an even number of inversions, and it is odd if it has an odd number of inversions. Prove that an n-cycle is even if n is odd, and odd if n is even....
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