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Unformatted text preview: MATH 239 Winter 2009 Assignment 3 Solutions [41 marks] 1. Let u k,n denote the number of solutions to the equation t 1 + ··· + t k = n where each t i is an integer, n is an integer, k ≥ 6, t i ≥ 10 for 1 ≤ i ≤ 6 and t i ≥ 0 for i > 6. Prove that X n ≥ u k,n x n = x 60 (1 x ) k . From this, find a simple formula for u k,n . Solution. [8 marks] For i ≤ 6 the possible values of t i are simply N ≥ 10 = { 10 , 11 , . . . } . As usual define w ( t i ) = t i . Then the generating function to use for t i , i ≤ 6, is Φ N ≥ 10 ( x ) = x 10 + x 11 + ··· = x 10 (1 + x + ··· ) = x 10 1 x . Similarly, for i ≥ 7, t i ∈ N ≥ ( x ) = { , 1 , . . . } and Φ N ≥ ( x ) = 1 + x + ··· = 1 1 x . As usual, define the weight of a ktuple w ( t 1 , . . . , t k ) to be t 1 + ··· + t k . Then u k,n is the number of ktuples of weight n such that the conditions on the t i hold, i.e. the ktuples in the set N 6 ≥ 10 × N k 6 ≥ . Since the weights we have defined have the property w ( t 1 , . . . , t k ) = ∑ w ( t i ), we have by the product rule for generating functions Φ N 6 ≥ 10 × N k 6 ≥ ( s ) = Φ N ≥ 10 ( s ) 6 · Φ N ≥ ( x ) k 6 = x 10 1 x 6 · 1 1 x k 6 = x 60 (1 x ) k Finally, by definition of the generating function, u k,n is the coefficient of x n in this gener ating function. In other words, X n ≥ u k,n x n = x 60 (1 x ) k as required. For the second part of the question, just use the binomial theorem for negative integer powers: u k,n = [ x n ] x 60 (1 x ) k = [ x n 60 ](1 x ) k = n 60 + k 1 n 60 = n 61 + k n 60 . This also equals ( n 61+ k k 1 ) . (These answers are only valid for k ≥ 6, but that is implicit in the question.) 2. (In this question and the following ones on this assignment, the weight of a composition of n is equal to the number n itself.) (a) Let Φ( x ) be the generating function for the compositions of n into k parts, where k is a specified number, and each part is either odd or divisible by 4. Show that Φ( x ) = x k (1 + x 2 + x 3 ) k (1 x 4 ) k ....
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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
 Math, Combinatorics

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