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Derangements

# Derangements - CSE 21 Notes on Derangements A derangement...

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CSE 21: Notes on Derangements (April 26, 2007) A derangement of [ n ] is a permutation which has no fixed point. In other words, in a derangement, for all i [ n ], i is not on position i . For example, for n = 3, all the derangements are: 231 , 312. Generally, the number of derangements is n k =0 ( - 1) k n ! k ! . The following is a proof from class (which is not in the textbook). Proof. Consider all the k -lists of [ n ], for k = 0 , 1 , 2 , · · · , n . k = 0 k = 1 k = 2 k = 3 k = 4 1 12 123 1234 2 13 124 1243 3 . . . . . . . . . 4 43 432 4321 Let L be the collection of all such lists; the empty list ∅ ∈ L is a special list. We define an operation (function) f on L as follows: For each list l ∈ L , 1. let n 1 be the smallest number that does not show up in l ; 2. let n 2 be the smallest number in l that is a fixed point; 3. If ( n 1 < n 2 ) or n 2 does not exist, insert n 1 into l on position n 1 (then n 1 is a fixed point), 4. If ( n 1 > n 2 ) or n 1 does not exist, remove n 2 from l . 5. If neither n 1 nor n 2 exist, do nothing.

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Derangements - CSE 21 Notes on Derangements A derangement...

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