Derangements¶
AUTHORS:
- Alasdair McAndrew (2010-05): Initial version
- Travis Scrimshaw (2013-03-30): Put derangements into category framework
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sage.combinat.derangements.
Derangement
¶ A derangement.
A derangement on a set \(S\) is a permutation \(\sigma\) such that \(\sigma(x) \neq x\) for all \(x \in S\), i.e. \(\sigma\) is a permutation of \(S\) with no fixed points.
EXAMPLES:
sage: D = Derangements(4) sage: elt = D([4,3,2,1]) sage: TestSuite(elt).run()
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sage.combinat.derangements.
Derangements
¶ The class of all derangements of a set or multiset.
A derangement on a set \(S\) is a permutation \(\sigma\) such that \(\sigma(x) \neq x\) for all \(x \in S\), i.e. \(\sigma\) is a permutation of \(S\) with no fixed points.
For an integer, or a list or string with all elements distinct, the derangements are obtained by a standard result described in [BV2004]. For a list or string with repeated elements, the derangements are formed by computing all permutations of the input and discarding all non-derangements.
INPUT:
x
– Can be an integer which corresponds to derangements of \(\{1, 2, 3, \ldots, x\}\), a list, or a string
REFERENCES:
EXAMPLES:
sage: D1 = Derangements([2,3,4,5]) sage: D1.list() [[3, 4, 5, 2], [5, 4, 2, 3], [3, 5, 2, 4], [4, 5, 3, 2], [4, 2, 5, 3], [5, 2, 3, 4], [5, 4, 3, 2], [4, 5, 2, 3], [3, 2, 5, 4]] sage: D1.cardinality() 9 sage: D1.random_element() # random [4, 2, 5, 3] sage: D2 = Derangements([1,2,3,1,2,3]) sage: D2.cardinality() 10 sage: D2.list() [[2, 1, 1, 3, 3, 2], [2, 1, 2, 3, 3, 1], [2, 3, 1, 2, 3, 1], [2, 3, 1, 3, 1, 2], [2, 3, 2, 3, 1, 1], [3, 1, 1, 2, 3, 2], [3, 1, 2, 2, 3, 1], [3, 1, 2, 3, 1, 2], [3, 3, 1, 2, 1, 2], [3, 3, 2, 2, 1, 1]] sage: D2.random_element() # random [2, 3, 1, 3, 1, 2]