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# hw7 - Math 146 Algebraic Combinatorics Homework 7 This...

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Math 146: Algebraic Combinatorics Homework 7 This problem set is due Friday, May 14. Do problems 3.11 and 3.12, in addition to the following: GK7.1. The aim of this exercise is yet another proof of the exponential formula. Let A ( x ) = k = 1 a k x k k ! be the exponential generating function for some types of animals of positive sizes. You’re suppose to prove everything in this exercise directly, rather than by using theorems in chapter 3. (a) Show that there are j + k + = n j + k + j , k , a j a k a l menageries of size n with three numbered animals. (b) Show that A ( x ) 3 is the e.g.f. for menageries with three numbered animals, more gener- ally that A ( x ) n is the e.g.f. for menageries with n numbered animals, and finally that A ( x ) n / n ! is the e.g.f. for menageries with n unnumbered animals. (c) Use part (b) to prove that M ( x ) = exp A ( x ) . GK7.2. Let f n be the number of partitions of [ n ] such that the number of parts is divisible by 4. Show (and the previous problem may be helpful) that the e.g.f. is F ( x ) = cos ( e x - 1 )+ cosh ( e x - 1 ) 2 .

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