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# PS03 - MATH 681 Problem Set#3 This problem set is due at...

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MATH 681 Problem Set #3 This problem set is due at the beginning of class on October 1 . 1. (10 points) Prove the following combinatorial identities: (a) (5 points) Recall that S ( n,k ) is equal to the number of ways to subdivide an n -element set into k nonempty parts. Produce a combinatorial argument to show that S ( n,k ) = kS ( n - 1 ,k ) + S ( n - 1 ,k - 1). (b) (5 points) Prove that for any m < n , m k =0 ( n k,m - k,n - m ) = 2 m ( n m ) . 2. (10 points) We know that p k ( n ) is the number of partitions of the number n into exactly k nonzero parts, and it has a generating function given by n =0 p k ( n ) x n = x k (1 - x )(1 - x 2 )(1 - x 3 )(1 - x 4 ) ··· (1 - x k ) . (a) (5 points) Prove that p k ( n ) = p k - 1 ( n - 1) + p k ( n - k ) by using a direct combi- natorial method (e.g. bijection or alternative enumerations of the same set). (b) (5 points) Prove that p k ( n ) = p k - 1 ( n - 1)+ p k ( n - k ) by equating the generating function n =0 p k ( n ) x n to the sum
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