a1 - A (where the weight of the empty set is dened to be...

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MATH 239 Assignment 1 This assignment is due at noon on Friday, May 15, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. Let integers 0 ` k n be given. Consider the binomial identity ± n k ²± k ` ² = ± n ` ²± n - ` k - ` ² . (a) Give an algebraic proof of this identity. (b) Give a combinatorial proof of this identity. 2. Let integers 0 ` k n be given. Consider the binomial identity ± n ` ² = ` X j =0 ± n - k j ²± k ` - j ² . (a) Give an algebraic proof of this identity. (Hint: use the binomial theorem.) (b) Give a combinatorial proof of this identity. 3. Let S be the set of all eight subsets of { 1 , 2 , 3 } . (a) Write down the generating function for S where the weight of a subset A is defined to be the number of elements in A . (b) Write down the generating function for S where the weight of a subset A is defined to be the largest element in
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Unformatted text preview: A (where the weight of the empty set is dened to be zero). (c) Write down the generating function for S where the weight of a subset A is dened to be the sum of the elements in A . 4. Let N = { , 1 , 2 , 3 , . . . } , and let the weight of n N be w ( n ) = n/ 4 if 4 | n, n/ 2 if n 2(mod 4) , n otherwise. Find the generating function N ( x ) in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials. 5. Find the inverse of the formal power series A ( x ) = X n n X k =0 20 k (-2) n-k x n . Write the inverse in the form p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials. (Hint: rst try to write A ( x ) as a product of two other formal power series.) 1...
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This note was uploaded on 10/23/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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