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.
class sage.combinat.perfect_matching.PerfectMatching(parent, s, check=True, sort=True)

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
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)
conjugate_by_permutation(*args, **kwds)

Deprecated: Use apply_permutation() instead. See trac ticket #23982 for details.

deprecated_function_alias(trac_number, func)

Create an aliased version of a function or a method which raises 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
is_non_crossing(*args, **kwds)

Deprecated: Use is_noncrossing() instead. See trac ticket #23982 for details.

is_non_nesting(*args, **kwds)

Deprecated: Use is_nonnesting() instead. See trac ticket #23982 for details.

loop_type(other=None)

Return the loop type of self.

INPUT:

• other – a perfect matching of the same set of self. (if the second argument is empty, the method an_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]
loops(other=None)

Return the loops of self.

INPUT:

• other – a perfect matching of the same set of self. (if the second argument is empty, the method an_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: loops = m.loops(n)
sage: loops # random
[['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]]
loops_iterator(other=None)

Iterate through the loops of self.

INPUT:

• other – a perfect matching of the same set of self. (if the second argument is empty, the method an_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
number_of_loops(other=None)

Return the number of loops of self.

INPUT:

• other – a perfect matching of the same set of self. (if the second argument is empty, the method an_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
partner(x)

Return the element in the same pair than x in the matching self.

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'
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)]
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)
[]
to_non_crossing_set_partition(*args, **kwds)

Deprecated: Use to_noncrossing_set_partition() instead. See trac ticket #23982 for details.

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()
{}
class sage.combinat.perfect_matching.PerfectMatchings(s)

Perfect matchings of a ground set.

INPUT:

• s – an iterable 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: x = M.an_element()
sage: x # random
[('a', 'c'), ('b', 'e'), ('d', 'f')]
sage: all(PerfectMatchings(i).an_element() in PerfectMatchings(i)
....:     for i in range(2,11,2))
True
Element

alias of PerfectMatching

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]
base_set()

Return the base set of self.

EXAMPLES:

sage: PerfectMatchings(3).base_set()
{1, 2, 3}
base_set_cardinality()

Return the cardinality of the base set of self.

EXAMPLES:

sage: PerfectMatchings(3).base_set_cardinality()
3
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
random_element()

Return a random element of self.

EXAMPLES:

sage: M = PerfectMatchings(('a', 'e', 'b', 'f', 'c', 'd'))
sage: x = M.random_element()
sage: x # random
[('a', 'b'), ('c', 'd'), ('e', 'f')]