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Unformatted text preview: MATH 239 Quiz No calculators or other aids may be used. Show all your work. 1. Let S = V × Y × Z where • V is the set of all positive odd integers • Y is the set { 4 , 8 , 12 , 16 ,..., 200 } • Z is the set of all nonnegative even integers. Find the generating function for S with respect to the weight function w defined by w ( v,y,z ) = v + ( y/ 4) + z . Express your answer in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials in x . Solution: • the generating function for V with respect to the weight function w ( v ) = v is x + x 3 + x 5 + ... = x (1 x 2 ) 1 . • the generating function for Y with respect to the weight function w ( y ) = y/ 4 is x + x 2 + x 3 + x 4 + ... + x 50 = x (1 + x + x 2 + ... + x 49 ) = x 1 x 50 1 x . • the generating function for Z with respect to the weight function w ( z ) = z is 1 + x 2 + x 4 + ... = (1 x 2 ) 1 . Since S is a Cartesian product, and the conditions for the Product Lemma are satisfied, we find the generating function for...
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 Spring '09
 M.PEI
 Combinatorics, Power Series, Integers, Cartesian product, Weight function, Formal power series

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