PS03-solutions - MATH 681 Problem Set #3 1. (10 points)...

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Unformatted text preview: MATH 681 Problem Set #3 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) . Let us consider the ways to divide { 1 , 2 ,...,n } into k nonempty sets. There are two possibilities to be addressed: either n is in a set by itself or with other numbers. If n is in a set by itself, then removal of this set { n } leaves us with a partition of the numbers { 1 , 2 ,...,n- 1 } into k- 1 sets. There are S ( n- 1 ,k- 1) such partitions, and each of them can be identified uniquely with a k-partition of { 1 , 2 ,...,n- 1 } by adding in the extra set { n } . On the other hand, if n is in a set with other numbers, removal of n from this set leaves k sets behind partitioning the numbers { 1 , 2 ,...,n- 1 } . There are S ( n- 1 ,k ) such possible partitions, but they are not uniquely identified with k-partitions of { 1 , 2 ,...,n- 1 } , since the term n could have been removed from (and can be reinserted into) any of the k sets. Thus, there are k different ways to build a k-partition of { 1 , 2 ,...,n } from a k-partition of { 1 , 2 ,...,n- 1 } , so this possibility accounts for k S ( n- 1 ,k ) partitions. Assembling these two cases, we see that S ( n,k ) = S ( n- 1 ,k- 1) + kS ( n- 1 ,k ). (b) (5 points) Prove that for any m < n , m k =0 ( n k,m- k,n- m ) = 2 m ( n m ) . This is actually very much like problem #1 from the previous problem set; its actually a generalization, since that problem is equivalent to the m = n- 1 case of this one. The left side is fairly easy to interpret: it represents the number of ways to divide an n-element set into three distinct parts of size k , m- k , and n- m , for any value of k . So, more properly, it represents the number of ways to divide n elements into classes A, B, and C so that classes A and B together contain m elements, and class C has n- m ....
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PS03-solutions - MATH 681 Problem Set #3 1. (10 points)...

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