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By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

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Additional info for Algorithms and Complexity (Second edition)

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Find the asymptotic behavior in the form xn ∼? 33). 8. 3. 9. 3 we find the phrase “... ” Prove that this phrase is justified, in that the equation shown always has exactly one positive real root. Exactly what special properties of that equation did you use in your proof? 5 Counting For a given positive integer n, consider the set {1, 2, . . n}. We will denote this set by the symbol [n], and we want to discuss the number of subsets of various kinds that it has. Here is a list of all of the subsets of [2]: ∅, {1}, {2}, {1, 2}.

The second element of the sequence can be any of the remaining n − 1 objects, so there are n(n − 1) possible ways to make the first two decisions. Then there are n − 2 choices for the third element, and so we have n(n − 1)(n − 2) ways to arrange the first three elements of the sequence. It is no doubt clear now that there are exactly n(n − 1)(n − 2) · · · 3 · 2 · 1 = n! ways to form the whole sequence. Of the 2n subsets of [n], how many have exactly k objects in them? The number of elements in a set is called its cardinality.

9, in the base 2 we need only 0 and 1, in the base 16 we need sixteen digits, etc. Proof. If b is fixed, the proof is by induction on n, the number being represented. ). Suppose, inductively, that every integer 1, 2, . . , n−1 is uniquely representable. Now consider the integer n. Define d = n mod b. Then d is one of the b permissible digits. By induction, the number n0 = (n − d)/b is uniquely representable, say: n−d = d0 + d1 b + d2 b2 + . . b Then clearly n n−d b b = d + d0 b + d1 b2 + d2 b3 + .

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Algorithms and Complexity (Second edition) by Herbert S. Wilf


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