qusol - MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We...

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Unformatted text preview: MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We have (2) S ( x ) = s s S x w 2 ( s ) = s s S x aw 1 ( s )+ b = s s S x b ( x a ) w 1 ( s ) = x b s s S ( x a ) w 1 ( s ) = x b (1) S ( x a ) , as required. 2. (a) Let N odd denote the set of odd positive integers, and N even denote the set of even non-negative 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 ] S ( x ). S = N k odd N k even , and the weight function condition for the Product Lemma is satisFed, so we have S ( x ) = ( N odd ( x )) k ( N even ( x )) k . Now, N odd ( x ) = x + x 3 + x 5 + . . . = x (1+ x 2 + x 4 + . . . ) = x 1-x 2 and N even ( x ) = 1+ x 2 + x 4 + . . . = 1 1-x 2 , so S ( x ) = ( x 1-x 2 ) k ( 1 1-x 2 ) k = x k (1-x 2 ) 2 k , as required. as required....
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qusol - MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We...

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