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hw13 - following analog of Pascal’s relations S n,k = S...

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Math 55: Discrete Mathematics, Fall 2008 Reading and Homework Assignment 13 Reading: Lecture 36: 8.5 Lecture 37: 5.5, Examples 10-11; 7.6, Theorem 1 and Examples 2-3 Homework (due Monday, 12/1): Self-checking problems 8.5: 3, 15, 21-22, 39, 45, 51 Problems to hand in: 8.5: 16, 24(a,b), 40, 46, 54 (A) Prove that the smallest equivalence relation containing a given relation R is the transitive closure S * of the reflexive and symmetric closure S = R R - 1 Δ. (B) Find an example of a relation R on a set with 3 elements such that the reflex- ive and symmetric closure of the transitive closure R * is not an equivalence relation. (C) Give a combinatorial proof that the Stirling numbers S ( n, k ) satisfy the
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Unformatted text preview: following analog of Pascal’s relations: S ( n,k ) = S ( n-1 ,k-1) + kS ( n-1 ,k ) for k,n > 0. [Hint: if X denotes the set of partitions of [ n ] into k parts, consider the decomposition X = A ∪ B , where A consists of partitions in which { n } is a block, and B consists of partitions in which n belongs to a block containing at least one other element.] Use the above relation and the initial conditions S ( n, 0) = 1 if n = 0, S ( n, 0) = 0 if n > 0 to make a table of S ( n,k ) for n and k less than or equal to 7. Check by comparing the value for S (7 , 3) in your table to the value given by the formula in Chapter 5.5....
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