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College Algebra Exam Review 11

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

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1.5. PERMUTATIONS 21 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 S n in cycle notation. Let a 1 be the first number ( 1 a 1 n ) which is not fixed by . Write a 2 D .a 1 / a 3 D .a 2 / D . .a 1 // a 4 D .a 3 / D . . .a 1 ///; and so forth. The numbers a 1 ; a 2 ; : : : cannot be all distinct since each is in f 1; 2; : : : ; n g . It follows that there is a number k such that a 1 ; a 2 ; : : : ; a k are all distinct, and .a k / D a 1 . (Ex- ercise 1.5.14 ). The permutation permutes the numbers f a 1 ; a 2 ; : : : ; a k g among themselves, and the remaining numbers f 1; 2; : : : ; n g n f a 1 ; a 2 ; : : : ; a k g among themselves, and the restriction of to f a 1 ; a 2 ; : : : ; a k g is the cy- cle .a 1 ; a 2 ; : : : ; a k / (Exercise 1.5.13 ). If fixes all numbers in f 1; 2; : : : ; n g n f a 1 ; a 2 ; : : : ; a k g , then D .a 1 ; a 2 ; : : : ; a k /: Otherwise, consider the first number b 1 62 f a 1 ; a 2 ; : : : ; a k g
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