This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Combinatorics, Integers

Click to edit the document details