4802Combinatorics07 - Math 4802 Fall 2007 Combinatorics...

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Unformatted text preview: Math 4802 Fall 2007 Combinatorics Problems (due Tuesday, October 30) 1. Define a selfish set to be a set which has its own cardinality (number of elements) as an element. A selfish set is called minimal if none of its proper subsets is selfish. Find the number of minimal selfish subsets of {1, 2, . . . , n}. 2. How many ways are there to select two disjoint subsets from {1, 2, . . . , n}? 1 1 3. Let Hn = 1+ 2 + · · · + n be the nth harmonic number, and write Hn = An . n! Show that An is the number of permutations of {1, 2, . . . , n + 1} having exactly two cycles. 4. Find a formula for the average number of cycles in a permutation of {1, 2, . . . , n}. 5. Given a finite set S of nonzero real numbers, let RS be the product of the reciprocals of all elements of S . Find a formula for the sum of RS over all non-empty subsets S of {1, 2, . . . , n}. 6. Given a collection S of 2n−1 subsets of {1, 2, . . . , n} such that any three elements of S have non-empty intersection, prove that the intersection of all the sets in S is non-empty. 1 ...
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This note was uploaded on 03/13/2010 for the course MATH 4801 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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