Unformatted text preview: n letters into a colored permutation on n1 letters, and some colored object for which there are 1 + x + x 2 + Â·Â·Â· + x n1 choices. (Hint: It looks scary, but it is a lot like the proof that the number of permutations is n !, which is the x = 1 case of the problem.) GK3.2. Prove that the ordinary generating function for Bell numbers, L ( x ) = âˆž âˆ‘ n = 1 b ( n ) x n , has 0 radius of convergence. (Hint: There are several ways to do this. One way is to show that b ( 2 n ) â‰¥ n ! with an explicit injection from permutations on n letters to partitions of 2 n letters.) *GK3.3. Find a counting proof of this fact for positive integers n and ` : ` âˆ‘ k = 1 Â± n k Â² ` ! ( `k ) ! = ` n . (Starred problems are extra credit, and I do not usually give hints for those.) 1...
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 Spring '10
 GregKuperberg
 Natural number, inversion generating function, weighted counting problem

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