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4203takehome - of n with precisely k inversions Prove that...

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Combinatorics (MAD 4203) — Fall 2010 Due Wednesday 12/8 Take-home Final Project If you choose to complete this take-home final project, it will be counted either as 100 midterm points (like a third midterm), or as 100 homework points, or not counted at all, whichever is most beneficial to you. Each question is worth 20 points. 1 How many n × n square matrices are there whose entries are each 0 or 1 and in which each row and column has an even sum? 2 Suppose we want to select as many subsets of [ n ] as possible such that every pair of subsets shares at least on element in common. How many subsets can we select? 3 Remember that an inversion in the permutation π is a pair of indices i and j with i < j so that π ( i ) > π ( j ). Let I ( n, k ) denote the number of permutations
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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 ) ··· (1 + x + ··· + x n-1 ) . 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 e-x/ 2-x 2 / 4 √ 1-x . 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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