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Unformatted text preview: 3.1. PERMUTATIONS 91 (d) Set the expression found in part (c) equal to- log(2), and solve for d as a function of n , thereby showing that d ∼ 2(log 2) n . Hint : If all three summands in the expression found in part (b) are used, one obtains a cubic equation in d . If the smallest of the three terms is thrown away, one obtains a quadratic equation in d . (e) Use a computer to calculate the exact values of d for various values of n . Compare these values with the approximate values obtained by using the answer to part d). 20 At a mathematical conference, ten participants are randomly seated around a circular table for meals. Using simulation, estimate the probability that no two people sit next to each other at both lunch and dinner. Can you make an intelligent conjecture for the case of n participants when n is large? 21 Modify the program AllPermutations to count the number of permutations of n objects that have exactly j fixed points for j = 0, 1, 2, . .. , n . Run your program for n = 2 to 6. Make a conjecture for the relation between the number that have 0 fixed points and the number that have exactly 1 fixed point. A proof of the correct conjecture can be found in Wilf....
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This note was uploaded on 09/15/2009 for the course SCF scf taught by Professor Scf during the Spring '09 term at Indian Institute Of Management, Ahmedabad.
- Spring '09