Perfect matchings¶
A perfect matching of a set \(S\) is a partition into 2-element sets. If \(S\) is the set \(\{1,...,n\}\), it is equivalent to fixpoint-free involutions. These simple combinatorial objects appear in different domains such as combinatoric of orthogonal polynomials and of the hyperoctaedral groups (see [MV], [McD] and also [CM]):
AUTHOR:
- Valentin Feray, 2010 : initial version
- Martin Rubey, 2017: inherit from SetPartition, move crossings and nestings to SetPartition
EXAMPLES:
Create a perfect matching:
sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]);m
[('a', 'e'), ('b', 'c'), ('d', 'f')]
Count its crossings, if the ground set is totally ordered:
sage: n = PerfectMatching([3,8,1,7,6,5,4,2]); n
[(1, 3), (2, 8), (4, 7), (5, 6)]
sage: n.number_of_crossings()
1
List the perfect matchings of a given ground set:
sage: PerfectMatchings(4).list()
[[(1, 2), (3, 4)], [(1, 3), (2, 4)], [(1, 4), (2, 3)]]
REFERENCES:
[MV] | combinatorics of orthogonal polynomials (A. de Medicis et X.Viennot, Moments des q-polynomes de Laguerre et la bijection de Foata-Zeilberger, Adv. Appl. Math., 15 (1994), 262-304) |
[McD] | combinatorics of hyperoctahedral group, double coset algebra and zonal polynomials (I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, second edition, 1995, chapter VII). |
[CM] | (1, 2, 3) Benoit Collins, Sho Matsumoto, On some properties of orthogonal Weingarten functions, Arxiv 0903.5143. |
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class
sage.combinat.perfect_matching.
PerfectMatching
(parent, s, check=True)¶ Bases:
sage.combinat.set_partition.SetPartition
A perfect matching.
A perfect matching of a set \(X\) is a set partition of \(X\) where all parts have size 2.
A perfect matching can be created from a list of pairs or from a fixed point-free involution as follows:
sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]);m [('a', 'e'), ('b', 'c'), ('d', 'f')] sage: n = PerfectMatching([3,8,1,7,6,5,4,2]);n [(1, 3), (2, 8), (4, 7), (5, 6)] sage: isinstance(m,PerfectMatching) True
The parent, which is the set of perfect matchings of the ground set, is automatically created:
sage: n.parent() Perfect matchings of {1, 2, 3, 4, 5, 6, 7, 8}
If the ground set is ordered, one can, for example, ask if the matching is non crossing:
sage: PerfectMatching([(1, 4), (2, 3), (5, 6)]).is_noncrossing() True
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Weingarten_function
(d, other=None)¶ Return the Weingarten function of two pairings.
This function is the value of some integrals over the orthogonal groups \(O_N\). With the convention of [CM], the method returns \(Wg^{O(d)}(other,self)\).
EXAMPLES:
sage: var('N') N sage: m = PerfectMatching([(1,3),(2,4)]) sage: n = PerfectMatching([(1,2),(3,4)]) sage: factor(m.Weingarten_function(N,n)) -1/((N + 2)*(N - 1)*N)
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conjugate_by_permutation
(*args, **kwds)¶ Deprecated: Use
apply_permutation()
instead. See trac ticket #23982 for details.
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deprecated_function_alias
(trac_number, func)¶ Create an aliased version of a function or a method which raise a deprecation warning message.
If f is a function or a method, write
g = deprecated_function_alias(trac_number, f)
to make a deprecated aliased version of f.INPUT:
trac_number
– integer. The trac ticket number where the deprecation is introduced.func
– the function or method to be aliased
EXAMPLES:
sage: from sage.misc.superseded import deprecated_function_alias sage: g = deprecated_function_alias(13109, number_of_partitions) sage: g(5) doctest:...: DeprecationWarning: g is deprecated. Please use sage.combinat.partition.number_of_partitions instead. See http://trac.sagemath.org/13109 for details. 7
This also works for methods:
sage: class cls(object): ....: def new_meth(self): return 42 ....: old_meth = deprecated_function_alias(13109, new_meth) sage: cls().old_meth() doctest:...: DeprecationWarning: old_meth is deprecated. Please use new_meth instead. See http://trac.sagemath.org/13109 for details. 42
sage: def a(): pass sage: b = deprecated_function_alias(13109, a) sage: b() doctest:...: DeprecationWarning: b is deprecated. Please use a instead. See http://trac.sagemath.org/13109 for details.
AUTHORS:
- Florent Hivert (2009-11-23), with the help of Mike Hansen.
- Luca De Feo (2011-07-11), printing the full module path when different from old path
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is_non_crossing
(*args, **kwds)¶ Deprecated: Use
is_noncrossing()
instead. See trac ticket #23982 for details.
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is_non_nesting
(*args, **kwds)¶ Deprecated: Use
is_nonnesting()
instead. See trac ticket #23982 for details.
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loop_type
(other=None)¶ Return the loop type of
self
.INPUT:
other
– a perfect matching of the same set ofself
. (if the second argument is empty, the methodan_element()
is called on the parent of the first)
OUTPUT:
If we draw the two perfect matchings simultaneously as edges of a graph, the graph obtained is a union of cycles of even lengths. The function returns the ordered list of the semi-length of these cycles (considered as a partition)
EXAMPLES:
sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]) sage: n = PerfectMatching([('a','b'),('d','f'),('e','c')]) sage: m.loop_type(n) [2, 1]
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loops
(other=None)¶ Return the loops of
self
.INPUT:
other
– a perfect matching of the same set ofself
. (if the second argument is empty, the methodan_element()
is called on the parent of the first)
OUTPUT:
If we draw the two perfect matchings simultaneously as edges of a graph, the graph obtained is a union of cycles of even lengths. The function returns the list of these cycles (each cycle is given as a list).
EXAMPLES:
sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]) sage: n = PerfectMatching([('a','b'),('d','f'),('e','c')]) sage: m.loops(n) [['a', 'e', 'c', 'b'], ['d', 'f']] sage: o = PerfectMatching([(1, 7), (2, 4), (3, 8), (5, 6)]) sage: p = PerfectMatching([(1, 6), (2, 7), (3, 4), (5, 8)]) sage: o.loops(p) [[1, 7, 2, 4, 3, 8, 5, 6]]
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loops_iterator
(other=None)¶ Iterate through the loops of
self
.INPUT:
other
– a perfect matching of the same set ofself
. (if the second argument is empty, the methodan_element()
is called on the parent of the first)
OUTPUT:
If we draw the two perfect matchings simultaneously as edges of a graph, the graph obtained is a union of cycles of even lengths. The function returns an iterator for these cycles (each cycle is given as a list).
EXAMPLES:
sage: o = PerfectMatching([(1, 7), (2, 4), (3, 8), (5, 6)]) sage: p = PerfectMatching([(1, 6), (2, 7), (3, 4), (5, 8)]) sage: it = o.loops_iterator(p) sage: next(it) [1, 7, 2, 4, 3, 8, 5, 6] sage: next(it) Traceback (most recent call last): ... StopIteration
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number_of_loops
(other=None)¶ Return the number of loops of
self
.INPUT:
other
– a perfect matching of the same set ofself
. (if the second argument is empty, the methodan_element()
is called on the parent of the first)
OUTPUT:
If we draw the two perfect matchings simultaneously as edges of a graph, the graph obtained is a union of cycles of even lengths. The function returns their numbers.
EXAMPLES:
sage: m = PerfectMatching([('a','e'),('b','c'),('d','f')]) sage: n = PerfectMatching([('a','b'),('d','f'),('e','c')]) sage: m.number_of_loops(n) 2
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partner
(x)¶ Return the element in the same pair than
x
in the matchingself
.EXAMPLES:
sage: m = PerfectMatching([(-3, 1), (2, 4), (-2, 7)]) sage: m.partner(4) 2 sage: n = PerfectMatching([('c','b'),('d','f'),('e','a')]) sage: n.partner('c') 'b'
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standardization
()¶ Return the standardization of
self
.See
SetPartition.standardization()
for details.EXAMPLES:
sage: n = PerfectMatching([('c','b'),('d','f'),('e','a')]) sage: n.standardization() [(1, 5), (2, 3), (4, 6)]
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to_graph
()¶ Return the graph corresponding to the perfect matching.
OUTPUT:
The realization of
self
as a graph.EXAMPLES:
sage: PerfectMatching([[1,3], [4,2]]).to_graph().edges(labels=False) [(1, 3), (2, 4)] sage: PerfectMatching([[1,4], [3,2]]).to_graph().edges(labels=False) [(1, 4), (2, 3)] sage: PerfectMatching([]).to_graph().edges(labels=False) []
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to_non_crossing_set_partition
(*args, **kwds)¶ Deprecated: Use
to_noncrossing_set_partition()
instead. See trac ticket #23982 for details.
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to_noncrossing_set_partition
()¶ Return the noncrossing set partition (on half as many elements) corresponding to the perfect matching if the perfect matching is noncrossing, and otherwise gives an error.
OUTPUT:
The realization of
self
as a noncrossing set partition.EXAMPLES:
sage: PerfectMatching([[1,3], [4,2]]).to_noncrossing_set_partition() Traceback (most recent call last): ... ValueError: matching must be non-crossing sage: PerfectMatching([[1,4], [3,2]]).to_noncrossing_set_partition() {{1, 2}} sage: PerfectMatching([]).to_noncrossing_set_partition() {}
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-
class
sage.combinat.perfect_matching.
PerfectMatchings
(s)¶ Bases:
sage.combinat.set_partition.SetPartitions_set
Perfect matchings of a ground set.
INPUT:
s
– an itegerable of hashable objects or an integer
EXAMPLES:
If the argument
s
is an integer \(n\), it will be transformed into the set \(\{1, \ldots, n\}\):sage: M = PerfectMatchings(6); M Perfect matchings of {1, 2, 3, 4, 5, 6} sage: PerfectMatchings([-1, -3, 1, 2]) Perfect matchings of {1, 2, -3, -1}
One can ask for the list, the cardinality or an element of a set of perfect matching:
sage: PerfectMatchings(4).list() [[(1, 2), (3, 4)], [(1, 3), (2, 4)], [(1, 4), (2, 3)]] sage: PerfectMatchings(8).cardinality() 105 sage: M = PerfectMatchings(('a', 'e', 'b', 'f', 'c', 'd')) sage: M.an_element() [('a', 'c'), ('b', 'e'), ('d', 'f')] sage: all(PerfectMatchings(i).an_element() in PerfectMatchings(i) ....: for i in range(2,11,2)) True
sage: S = PerfectMatchings(4) sage: elt = S([[1,3],[2,4]]); elt [(1, 3), (2, 4)] sage: S = PerfectMatchings([]) sage: S([]) []
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Element
¶ alias of
PerfectMatching
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Weingarten_matrix
(N)¶ Return the Weingarten matrix corresponding to the set of PerfectMatchings
self
.It is a useful theoretical tool to compute polynomial integral over the orthogonal group \(O_N\) (see [CM]).
EXAMPLES:
sage: M = PerfectMatchings(4).Weingarten_matrix(var('N')) sage: N*(N-1)*(N+2)*M.apply_map(factor) [N + 1 -1 -1] [ -1 N + 1 -1] [ -1 -1 N + 1]
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base_set
()¶ Return the base set of
self
.EXAMPLES:
sage: PerfectMatchings(3).base_set() {1, 2, 3}
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base_set_cardinality
()¶ Return the cardinality of the base set of
self
.EXAMPLES:
sage: PerfectMatchings(3).base_set_cardinality() 3
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cardinality
()¶ Return the cardinality of the set of perfect matchings
self
.This is \(1*3*5*...*(2n-1)\), where \(2n\) is the size of the ground set.
EXAMPLES:
sage: PerfectMatchings(8).cardinality() 105 sage: PerfectMatchings([1,2,3,4]).cardinality() 3 sage: PerfectMatchings(3).cardinality() 0 sage: PerfectMatchings([]).cardinality() 1
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random_element
()¶ Return a random element of
self
.EXAMPLES:
sage: M = PerfectMatchings(('a', 'e', 'b', 'f', 'c', 'd')) sage: M.random_element() [('a', 'b'), ('c', 'd'), ('e', 'f')]