Unformatted text preview: of [ n ] with precisely k inversions. Prove that ( n 2 ) s k =0 I ( n, k ) x k = (1 + x )(1 + x + x 2 )(1 + x + x 2 + x 3 ) Â·Â·Â· (1 + x + Â·Â·Â· + x n1 ) . 4 Let G be a simple graph on the vertices [ n ] in which each vertex has degree 2. Prove that G is a union of disjoint cycles. 5 Let g ( n ) denote the number of graphs on the vertices [ n ] which are unions of disjoint cycles. a. Prove that the exponential generating function for g ( n ) is ex/ 2x 2 / 4 âˆš 1x . b. Explain why the generating function in part (a) is diFerent from the exponential generating function for n !, even though permutations are also unions of disjoint cycles on the set [ n ]....
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 Fall '10
 Vetter
 Set Theory, Graph Theory, Generating function, exponential generating function

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