College Algebra Exam Review 11

College Algebra Exam Review 11 - 21 1.5. PERMUTATIONS...

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Unformatted text preview: 21 1.5. PERMUTATIONS shuffle into a product of disjoint cycles. (We cannot help but be curious about the order of the perfect shuffle of a standard deck of 52 cards.)7 Here is (the outline of) an algorithm for writing a permutation 2 Sn in cycle notation. Let a1 be the first number (1 Ä a1 Ä n) which is not fixed by . Write a2 D .a1 / a3 D .a2 / D . .a1 // a4 D .a3 / D . . .a1 ///; and so forth. The numbers a1 ; a2 ; : : : cannot be all distinct since each is in f1; 2; : : : ; ng. It follows that there is a number k such that a1 ; a2 ; : : : ; ak are all distinct, and .ak / D a1 . (Exercise 1.5.14). The permutation permutes the numbers fa1 ; a2 ; : : : ; ak g among themselves, and the remaining numbers f1; 2; : : : ; ng n fa1 ; a2 ; : : : ; ak g among themselves, and the restriction of to fa1 ; a2 ; : : : ; ak g is the cycle .a1 ; a2 ; : : : ; ak / (Exercise 1.5.13). If fixes all numbers in f1; 2; : : : ; ng n fa1 ; a2 ; : : : ; ak g, then D .a1 ; a2 ; : : : ; ak /: Otherwise, consider the first number b1 62 fa1 ; a2 ; : : : ; ak g that is not fixed by . Write b2 D .b1 / b3 D .b2 / D . .b1 // b4 D .b3 / D . . .b1 ///; and so forth; as before, there is an integer l such that b1 ; : : : ; bl are all distinct and .bl / D b1 . Now permutes the numbers fa1 ; a2 ; : : : ; ak g [ fb1 ; : : : ; bl g among themselves, and the remaining numbers f1; 2; : : : ; ng n .fa1 ; a2 ; : : : ; ak g [ fb1 ; : : : ; bl g/ among themselves; furthermore, the restriction of to fa1 ; a2 ; : : : ; ak g [ fb1 ; : : : ; bl g is the product of disjoint cycles .a1 ; a2 ; : : : ; ak /.b1 ; : : : ; bl /: 7 A sophisticated analysis of the mathematics of card shuffling is carried out in D. Aldous and P. Diaconis, “Shuffling cards and stopping times,” Amer. Math. Monthly, 93 (1986), no. 5, 333–348. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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