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Unformatted text preview: MATH 239 Quiz (Wednesday)  Suggested Solutions 1. We have Φ S × T ( x ) = s ( σ,τ ) ∈ S × T x w 2 ( σ,τ ) = s ( σ,τ ) ∈ S × T x w 1 ( σ ) = s σ ∈ S s τ ∈ T x w 1 ( σ ) = s σ ∈ S x w 1 ( σ ) s τ ∈ T 1 = p s σ ∈ S x w 1 ( σ ) P ·  T  = Φ S ( x ) ·  T  as required. 2. (a) Let N odd denote the set of odd positive integers, and N even > denote the set of even positive integers. Let S be the set of ordered lists ( t 1 , . . . t 2 k ) where t 1 , . . . , t k ∈ N odd and t k +1 , . . . , t 2 k ∈ N even > , and let w be given by w ( t 1 , . . . t 2 k ) = t 1 + . . . + t 2 k . Observe that a n is the number of conFgurations in S of weight n , so a n = [ x n ]Φ S ( x ). Then S = ( N odd ) k × ( N even > ) k , and the weight function condition for the Product Lemma is satisFed, so we have Φ S ( x ) = (Φ N odd ( x )) k (Φ N even > ( x )) k . Now, Φ N odd ( x ) = x + x 3 + x 5 + . . . = x (1+ x 2 + x 4 + . . . ) = x 1x 2 and Φ N even > ( x ) = x 2...
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This note was uploaded on 07/28/2009 for the course MATH math 239 taught by Professor .... during the Spring '08 term at Waterloo.
 Spring '08
 ....
 Integers

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