finalf04

# finalf04 - MATH 239 Final Exam, Dec. 20, 2004 2 1. (a) [2...

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MATH 239 Final Exam, Dec. 20, 2004 2 1. (a) [2 marks] List all compositions of 2, 3 and 4 where each part is a positive integer not equal to 2. (b) [2 marks] Prove that the generating function for the number of compositions of a positive integer n into k parts, such that each part is a positive integer which is not equal to two, is Φ( x ) = ± x - x 2 + x 3 1 - x ² k . (c) [2 marks] Deteremine the generating function for the number, a n , of compositions of n with no restriction on the number of parts in the composition. As in parts (a) and (b), each part is a positive integer which is not equal to 2.

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MATH 239 Final Exam, Dec. 20, 2004 3 (d) [3 marks] Let { b n } be the sequence whose generating function is deﬁned by X n 0 b n x n = 1 + x 4 1 - x - 2 x 3 . Determine a recurrence relation, together with suﬃcient initial conditions, to uniquely determine the sequence of b n ’s. (e) [3 marks] The recurrence relation c n = 5 c n - 1 - 6 c n - 2 for all integers n 2 with initial conditions c 0 = - 3 and c 1 = - 4, deﬁnes the terms of a sequence { c n } . Solve this recurrence relation.
MATH 239 Final Exam, Dec. 20, 2004 4 2. (a) For each of the following sets, write down a decomposition that uniquely creates the elements of that set. i. [2 marks] The set of { 0 , 1 } -strings where all blocks have even length. ii.

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## This note was uploaded on 07/28/2009 for the course MATH math239 taught by Professor ... during the Spring '06 term at Waterloo.

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finalf04 - MATH 239 Final Exam, Dec. 20, 2004 2 1. (a) [2...

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