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Unformatted text preview: MATH 239 Assignment 2 [Total 50 marks] This assignment is due at noon on Friday, May 29, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Determine the following coeﬃcient as a summation, where ` and m are nonnegative integers: [ x n ](1 + 2 x )` (1x 3 )m Solution: [ x n ](1 + 2 x )` (1x 3 )m = [ x n ] X k ≥ ± k + `1 `1 ² (2 x ) k X j ≥ ± j + m1 m1 ² x 3 j = [ x n ] X k ≥ X j ≥ ± k + `1 `1 ²± j + m1 m1 ² (2) k x k +3 j = X k +3 j = n ± k + `1 `1 ²± j + m1 m1 ² (2) k = b n 3 c X j =0 ± n3 j + `1 `1 ²± j + m1 m1 ² (2) n3 j 2. Find the generating function for the number of compositions of n into an even number of parts, where each part is odd. Solution: Let A = { 1 , 3 , 5 ,... } . Using the weight function given by w ( n ) = n for all n ∈ A , the generating function for A is Φ A ( x ) = x w (1) + x w (3) + x w (5) + ··· = x 1 + x 3 + x 5 + ··· = x (1 + x 2 + x 4 + ··· ) = x 1x 2 . The set S of all compositions having an even number of parts, each of which is odd, is S = A ∪ A 2 ∪ A 4 ∪ A 6 ∪ ··· By the Sum Lemma and the Product Lemma, Φ S ( x ) = Φ A ( x ) + Φ A 2 ( x ) + Φ A 4 ( x ) + ··· = ± x 1x 2 ² + ± x 1x 2 ² 2 + ± x 1x 2 ² 4 + ··· 1 Since this is a geometric series, and x 1x 2 has zero constant term, we see that...
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 Spring '09
 M.PEI
 Combinatorics, Integers, Recurrence relation, Generating function, Xn

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