hw3 - n letters into a colored permutation on n-1 letters...

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Math 146: Algebraic Combinatorics Homework 3 This problem set is due Friday, April 16. Do problems 1.16, 1.17, 1.21(a), 2.1(c), 2.2(a,d), and 2.11(a,d,e), in addition to my problems. In problem 1.17, all you are supposed to do is guess the generating function by factoring it in a few cases. One of my problems gives away the answer and also argues for a specific proof. GK3.1. Inversion numbers are a good excuse to interpret a generating polynomial as the answer to a weighted counting problem. Consider the inversion generating function B n ( x ) in problem 1.17, and let’s say that x is a positive integer and that you have a paintbox with x colors. Consider the problem of counting colored permutations, where each inversion of a permuta- tion σ is colored with one of the x colors, for a total of x k colorings of a permutation with k inversions. Prove the formula B n ( x ) = ( 1 + x + x 2 + ··· + x n - 1 ) B n - 1 ( x ) with a direct counting argument. Namely, decompose a colored permutation on
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Unformatted text preview: n letters into a colored permutation on n-1 letters, and some colored object for which there are 1 + x + x 2 + ··· + x n-1 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 ) = ∞ ∑ 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 ) ≥ 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 ` : ` ∑ k = 1 ± n k ² ` ! ( `-k ) ! = ` n . (Starred problems are extra credit, and I do not usually give hints for those.) 1...
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