hw2 - proof The idea of the proof is the “stars and...

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Math 146: Algebraic Combinatorics Homework 2 This problem set is due Friday, April 9. Do problems 1.6(a,c), 1.7, 1.8(a,c), and 1.11, in addition to the following: GK2.1. Express the generating functions in 1.6(a,c) using partial fractions, and then use that to solve the recurrences. GK2.2. The stars and bars construction. In class I will have discussed the multichoose coefficient n k , which to review means the number of ways to choose k things from set of n things, with repetitions allowed. Consider again the identity n k = n + k - 1 k . (1) We have a generating function proof of this identity; the point of this problem is a direct
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Unformatted text preview: proof. The idea of the proof is the “stars and bars” notation for a multichoice. The notation is that, if you are choosing elements from [ n ] , you should list a star for each time you choose 1, then a separator bar, then a star for each time you choose 2, then a separator, and so on. For example, if you choose { 1 , 1 , 2 , 4 } from [ 4 ] , then in stars and bars notation it is ?? | ? || ? . Prove that the elements of (( n k )) can be expressed with a string of n-1 bars and k stars, and then use that to prove the identity (1). 1...
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