Derangements#

AUTHORS:

Alasdair McAndrew (2010-05): Initial version
Travis Scrimshaw (2013-03-30): Put derangements into category framework

class sage.combinat.derangements.Derangement(parent, *args, **kwds)#

Bases: sage.combinat.combinat.CombinatorialElement

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()

to_permutation()#

Return the permutation corresponding to self.

EXAMPLES:

sage: D = Derangements(4)
sage: p = D([4,3,2,1]).to_permutation(); p
[4, 3, 2, 1]
sage: type(p)
<class 'sage.combinat.permutation.StandardPermutations_all_with_category.element_class'>
sage: D = Derangements([1, 3, 3, 4])
sage: D[0].to_permutation()
Traceback (most recent call last):
...
ValueError: Can only convert to a permutation for derangements of [1, 2, ..., n]

class sage.combinat.derangements.Derangements(x)#

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

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]

Element#: alias of Derangement

cardinality()#

Counts the number of derangements of a positive integer, a list, or a string. The list or string may contain repeated elements. If an integer \(n\) is given, the value returned is the number of derangements of \([1, 2, 3, \ldots, n]\).

For an integer, or a list or string with all elements distinct, the value is obtained by the standard result \(D_2 = 1, D_3 = 2, D_n = (n-1) (D_{n-1} + D_{n-2})\).

For a list or string with repeated elements, the number of derangements is computed by Macmahon’s theorem. If the numbers of repeated elements are \(a_1, a_2, \ldots, a_k\) then the number of derangements is given by the coefficient of \(x_1 x_2 \cdots x_k\) in the expansion of \(\prod_{i=0}^k (S - s_i)^{a_i}\) where \(S = x_1 + x_2 + \cdots + x_k\).

EXAMPLES:

sage: D = Derangements(5)
sage: D.cardinality()
44
sage: D = Derangements([1,44,918,67,254])
sage: D.cardinality()
44
sage: D = Derangements(['A','AT','CAT','CATS','CARTS'])
sage: D.cardinality()
44
sage: D = Derangements('UNCOPYRIGHTABLE')
sage: D.cardinality()
481066515734
sage: D = Derangements([1,1,2,2,3,3])
sage: D.cardinality()
10
sage: D = Derangements('SATTAS')
sage: D.cardinality()
10
sage: D = Derangements([1,1,2,2,2])
sage: D.cardinality()
0

random_element()#

Produces all derangements of a positive integer, a list, or a string. The list or string may contain repeated elements. If an integer \(n\) is given, then a random derangements of \([1, 2, 3, \ldots, n]\) is returned

For an integer, or a list or string with all elements distinct, the value is obtained by an algorithm described in [MPP2008]. For a list or string with repeated elements the derangement is formed by choosing an element at random from the list of all possible derangements.

OUTPUT:

A single list or string containing a derangement, or an empty list if there are no derangements.

EXAMPLES:

sage: D = Derangements(4)
sage: D.random_element() # random
[2, 3, 4, 1]
sage: D = Derangements(['A','AT','CAT','CATS','CARTS','CARETS'])
sage: D.random_element() # random
['AT', 'CARTS', 'A', 'CAT', 'CARETS', 'CATS']
sage: D = Derangements('UNCOPYRIGHTABLE')
sage: D.random_element() # random
['C', 'U', 'I', 'H', 'O', 'G', 'N', 'B', 'E', 'L', 'A', 'R', 'P', 'Y', 'T']
sage: D = Derangements([1,1,1,1,2,2,2,2,3,3,3,3])
sage: D.random_element() # random
[3, 2, 2, 3, 1, 3, 1, 3, 2, 1, 1, 2]
sage: D = Derangements('ESSENCES')
sage: D.random_element() # random
['N', 'E', 'E', 'C', 'S', 'S', 'S', 'E']
sage: D = Derangements([1,1,2,2,2])
sage: D.random_element()
[]