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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Natural number, Recurrence relation, Fibonacci number