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 n1 ( x ) + P n1 ( 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 ncycle is even if n is odd, and odd if n is even....
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 Fall '10
 Vetter
 Group Theory, Asymptotic Bounds

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