Constellations

A constellation is a tuple \((g_0, g_1, \dots, g_k)\) of permutations such that the product \(g_0 g_1 ... g_k\) is the identity. One often assumes that the group generated by \(g_0, g_1, \dots, g_k\) acts transitively ([LZ2004] definition 1). Geometrically, it corresponds to a covering of the 2-sphere ramified over \(k\) points (the transitivity condition corresponds to the connectivity of the covering).

EXAMPLES:

sage: c = Constellation(['(1,2)', '(1,3)', None])
sage: c
Constellation of length 3 and degree 3
g0 (1,2)(3)
g1 (1,3)(2)
g2 (1,3,2)
sage: C = Constellations(3,4); C
Connected constellations of length 3 and degree 4 on {1, 2, 3, 4}
sage: C.cardinality()  # long time
426

sage: C = Constellations(3, 4, domain=('a', 'b', 'c', 'd'))
sage: C
Connected constellations of length 3 and degree 4 on {'a', 'b', 'c', 'd'}
sage: c = C(('a','c'),(('b','c'),('a','d')), None)
sage: c
Constellation of length 3 and degree 4
g0 ('a','c')('b')('d')
g1 ('a','d')('b','c')
g2 ('a','d','c','b')
sage: c.is_connected()
True
sage: c.euler_characteristic()
2
sage: TestSuite(C).run()

sage.combinat.constellation.Constellation(g=None, mutable=False, connected=True, check=True)

Constellation

INPUT:

• g – a list of permutations

• mutable – whether the result is mutable or not. Default is False.

• connected – whether the result should be connected. Default is True.

• check – whether or not to check. If it is True, then the list g must contains no None.

EXAMPLES:

Simple initialization:

sage: Constellation(['(0,1)','(0,3)(1,2)','(0,3,1,2)'])
Constellation of length 3 and degree 4
g0 (0,1)(2)(3)
g1 (0,3)(1,2)
g2 (0,3,1,2)

One of the permutation can be omitted:

sage: Constellation(['(0,1)', None, '(0,4)(1,2,3)'])
Constellation of length 3 and degree 5
g0 (0,1)(2)(3)(4)
g1 (0,3,2,1,4)
g2 (0,4)(1,2,3)

One can define mutable constellations:

sage: Constellation(([0,2,1], [2,1,0], [1,2,0]), mutable=True)
Constellation of length 3 and degree 3
g0 (0)(1,2)
g1 (0,2)(1)
g2 (0,1,2)

class sage.combinat.constellation.Constellation_class(parent, g, connected, mutable, check)

Bases: sage.structure.element.Element

Constellation

A constellation or a tuple of permutations \((g_0,g_1,...,g_k)\) such that the product \(g_0 g_1 ... g_k\) is the identity.

braid_group_action(i)

Act on self as the braid group generator that exchanges position \(i\) and \(i+1\).

INPUT:

• i – integer in \([0, n-1]\) where \(n\) is the length of self

EXAMPLES:

sage: sigma = lambda c, i: c.braid_group_action(i)

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)',None]); c
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,4)(2)(3)
g2 (0,1,3,2,4)
sage: sigma(c, 1)
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0,1,3,2,4)
g2 (0,3)(1)(2)(4)

Check the commutation relation:

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)','(2,5)(0,4)',None])
sage: d = Constellation(['(0,1,3,5)','(2,3,4)','(0,3,5)',None])
sage: c13 = sigma(sigma(c, 0), 2)
sage: c31 = sigma(sigma(c, 2), 0)
sage: c13 == c31
True
sage: d13 = sigma(sigma(d, 0), 2)
sage: d31 = sigma(sigma(d, 2), 0)
sage: d13 == d31
True

Check the braid relation:

sage: c121 = sigma(sigma(sigma(c, 1), 2), 1)
sage: c212 = sigma(sigma(sigma(c, 2), 1), 2)
sage: c121 == c212
True
sage: d121 = sigma(sigma(sigma(d, 1), 2), 1)
sage: d212 = sigma(sigma(sigma(d, 2), 1), 2)
sage: d121 == d212
True

braid_group_orbit()

Return the graph of the action of the braid group.

The action is considered up to isomorphism of constellation.

EXAMPLES:

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)',None]); c
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,4)(2)(3)
g2 (0,1,3,2,4)
sage: G = c.braid_group_orbit()
sage: G.num_verts()
4
sage: G.num_edges()
12

connected_components()

Return the connected components.

OUTPUT:

A list of connected constellations.

EXAMPLES:

sage: c = Constellation(['(0,1)(2)', None, '(0,1)(2)'], connected=False)
sage: cc = c.connected_components(); cc
[Constellation of length 3 and degree 2
g0 (0,1)
g1 (0)(1)
g2 (0,1),
Constellation of length 3 and degree 1
g0 (0)
g1 (0)
g2 (0)]
sage: all(c2.is_connected() for c2 in cc)
True

sage: c = Constellation(['(0,1,2)', None], connected=False)
sage: c.connected_components()
[Constellation of length 2 and degree 3
g0 (0,1,2)
g1 (0,2,1)]

copy()

Return a copy of self.

degree()

Return the degree of the constellation.

The degree of a constellation is the number \(n\) that corresponds to the symmetric group \(S(n)\) in which the permutations of the constellation are defined.

EXAMPLES:

sage: c = Constellation([])
sage: c.degree()
0
sage: c = Constellation(['(0,1)',None])
sage: c.degree()
2
sage: c = Constellation(['(0,1)','(0,3,2)(1,5)',None,'(4,3,2,1)'])
sage: c.degree()
6

euler_characteristic()

Return the Euler characteristic of the surface.

ALGORITHM:

Hurwitz formula

EXAMPLES:

sage: c = Constellation(['(0,1)', '(0,2)', None])
sage: c.euler_characteristic()
2

sage: c = Constellation(['(0,1,2,3)','(1,3,0,2)', '(0,3,1,2)', None])
sage: c.euler_characteristic()
-4

g(i=None)

Return the permutation \(g_i\) of the constellation.

INPUT:

• i – integer or None (default)

If None , return instead the list of all \(g_i\).

EXAMPLES:

sage: c = Constellation(['(0,1,2)(3,4)','(0,3)',None])
sage: c.g(0)
(0,1,2)(3,4)
sage: c.g(1)
(0,3)
sage: c.g(2)
(0,4,3,2,1)
sage: c.g()
[(0,1,2)(3,4), (0,3), (0,4,3,2,1)]

genus()

Return the genus of the surface.

EXAMPLES:

sage: c = Constellation(['(0,1)', '(0,2)', None])
sage: c.genus()
0

sage: c = Constellation(['(0,1)(2,3,4)','(1,3,4)(2,0)', None])
sage: c.genus()
1

is_connected()

Test of connectedness.

EXAMPLES:

sage: c = Constellation(['(0,1)(2)', None, '(0,1)(2)'], connected=False)
sage: c.is_connected()
False
sage: c = Constellation(['(0,1,2)', None], connected=False)
sage: c.is_connected()
True

is_isomorphic(other, return_map=False)

Test of isomorphism.

Return True if the constellations are isomorphic (i.e. related by a common conjugacy) and return the permutation that conjugate the two permutations if return_map is True in such a way that self.relabel(m) == other.

ALGORITHM:

uses canonical labels obtained from the method relabel().

EXAMPLES:

sage: c = Constellation([[1,0,2],[2,1,0],[0,2,1],None])
sage: d = Constellation([[2,1,0],[0,2,1],[1,0,2],None])
sage: answer, mapping = c.is_isomorphic(d,return_map=True)
sage: answer
True
sage: c.relabel(mapping) == d
True

is_mutable()

Return False as self is immutable.

EXAMPLES:

sage: c = Constellation(([0,2,1],[2,1,0],[1,2,0]), mutable=False)
sage: c.is_mutable()
False

length()

Return the number of permutations.

EXAMPLES:

sage: c = Constellation(['(0,1)','(0,2)','(0,3)',None])
sage: c.length()
4
sage: c = Constellation(['(0,1,3)',None,'(1,2)'])
sage: c.length()
3

mutable_copy()

Return a mutable copy of self.

EXAMPLES:

sage: c = Constellation(([0,2,1],[2,1,0],[1,2,0]), mutable=False)
sage: d = c.mutable_copy()
sage: d.is_mutable()
True

passport(i=None)

Return the profile of self.

The profile of a constellation is the tuple of partitions associated to the conjugacy classes of the permutations of the constellation.

This is also called the passport.

EXAMPLES:

sage: c = Constellation(['(0,1,2)(3,4)','(0,3)',None])
sage: c.profile()
([3, 2], [2, 1, 1, 1], [5])

profile(i=None)

Return the profile of self.

The profile of a constellation is the tuple of partitions associated to the conjugacy classes of the permutations of the constellation.

This is also called the passport.

EXAMPLES:

sage: c = Constellation(['(0,1,2)(3,4)','(0,3)',None])
sage: c.profile()
([3, 2], [2, 1, 1, 1], [5])

relabel(perm=None, return_map=False)

Relabel self.

If perm is provided then relabel with respect to perm. Otherwise use canonical labels. In that case, if return_map is provided, the return also the map used for canonical labels.

Algorithm:

the cycle for g(0) are adjacent and the cycle are joined with respect to the other permutations. The minimum is taken for all possible renumerotations.

EXAMPLES:

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)',None]); c
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,4)(2)(3)
g2 (0,1,3,2,4)
sage: c2 = c.relabel(); c2
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,2)(3)(4)
g2 (0,1,4,3,2)

The map returned when the option return_map is set to True can be used to set the relabelling:

sage: c3, perm = c.relabel(return_map=True)
sage: c3 == c2 and c3 == c.relabel(perm=perm)
True

sage: S5 = SymmetricGroup(range(5))
sage: d = c.relabel(S5([4,3,1,0,2])); d
Constellation of length 3 and degree 5
g0 (0,2,1)(3,4)
g1 (0)(1)(2,3)(4)
g2 (0,1,2,4,3)
sage: d.is_isomorphic(c)
True

We check that after a random relabelling the new constellation is isomorphic to the initial one:

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)',None])
sage: p = S5.random_element()
sage: cc = c.relabel(perm=p)
sage: cc.is_isomorphic(c)
True

Check that it works for “non standard” labels:

sage: c = Constellation([(('a','b'),('c','d','e')),('b','d'), None])
sage: c.relabel()
Constellation of length 3 and degree 5
g0 ('a','b')('c','d','e')
g1 ('a')('b','c')('d')('e')
g2 ('a','b','e','d','c')

set_immutable()

Do nothing, as self is already immutable.

EXAMPLES:

sage: c = Constellation(([0,2,1],[2,1,0],[1,2,0]), mutable=False)
sage: c.set_immutable()
sage: c.is_mutable()
False

switch(i, j0, j1)

Perform the multiplication by the transposition \((j0, j1)\) between the permutations \(g_i\) and \(g_{i+1}\).

The modification is local in the sense that it modifies \(g_i\) and \(g_{i+1}\) but does not modify the product \(g_i g_{i+1}\). The new constellation is

\[(g_0, \ldots, g_{i-1}, g_{i} (j0 j1), (j0 j1) g_{i+1}, g_{i+2}, \ldots, g_k)\]

EXAMPLES:

sage: c = Constellation(['(0,1)(2,3,4)','(1,4)',None], mutable=True); c
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,4)(2)(3)
g2 (0,1,3,2,4)
sage: c.is_mutable()
True
sage: c.switch(1,2,3); c
Constellation of length 3 and degree 5
g0 (0,1)(2,3,4)
g1 (0)(1,4)(2,3)
g2 (0,1,3,4)(2)
sage: c._check()
sage: c.switch(2,1,3); c
Constellation of length 3 and degree 5
g0 (0,1,4,2,3)
g1 (0)(1,4)(2,3)
g2 (0,3,4)(1)(2)
sage: c._check()
sage: c.switch(0,0,1); c
Constellation of length 3 and degree 5
g0 (0)(1,4,2,3)
g1 (0,4,1)(2,3)
g2 (0,3,4)(1)(2)
sage: c._check()

sage.combinat.constellation.Constellations(*data, **options)

Build a set of constellations.

INPUT:

• profile – an optional profile

• length – an optional length

• degree – an optional degree

• connected – an optional boolean

EXAMPLES:

sage: Constellations(4,2)
Connected constellations of length 4 and degree 2 on {1, 2}

sage: Constellations([[3,2,1],[3,3],[3,3]])
Connected constellations with profile ([3, 2, 1], [3, 3], [3, 3]) on {1, 2, 3, 4, 5, 6}

class sage.combinat.constellation.Constellations_ld(length, degree, sym=None, connected=True)

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

Constellations of given length and degree.

EXAMPLES:

sage: C = Constellations(2,3); C
Connected constellations of length 2 and degree 3 on {1, 2, 3}
sage: C([[2,3,1],[3,1,2]])
Constellation of length 2 and degree 3
g0 (1,2,3)
g1 (1,3,2)
sage: C.cardinality()
2
sage: Constellations(2,3,connected=False).cardinality()
6

Element

alias of Constellation_class

braid_group_action()

Return a list of graphs that corresponds to the braid group action on self up to isomorphism.

OUTPUT:

• list of graphs

EXAMPLES:

sage: C = Constellations(3,3)
sage: C.braid_group_action()
[Looped multi-digraph on 3 vertices,
 Looped multi-digraph on 1 vertex,
 Looped multi-digraph on 3 vertices]

braid_group_orbits()

Return the orbits under the action of braid group.

EXAMPLES:

sage: C = Constellations(3,3)
sage: O = C.braid_group_orbits()
sage: len(O)
3
sage: [x.profile() for x in O[0]]
[([1, 1, 1], [3], [3]), ([3], [1, 1, 1], [3]), ([3], [3], [1, 1, 1])]
sage: [x.profile() for x in O[1]]
[([3], [3], [3])]
sage: [x.profile() for x in O[2]]
[([2, 1], [2, 1], [3]), ([2, 1], [3], [2, 1]), ([3], [2, 1], [2, 1])]

is_empty()

Return whether this set of constellations is empty.

EXAMPLES:

sage: Constellations(2, 3).is_empty()
False
sage: Constellations(1, 2).is_empty()
True
sage: Constellations(1, 2, connected=False).is_empty()
False

random_element(mutable=False)

Return a random element.

This is found by trial and rejection, starting from a random list of permutations.

EXAMPLES:

sage: const = Constellations(3,3)
sage: const.random_element()
Constellation of length 3 and degree 3
...
...
...
sage: c = const.random_element()
sage: c.degree() == 3 and c.length() == 3
True

class sage.combinat.constellation.Constellations_p(profile, domain=None, connected=True)

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

Constellations with fixed profile.

EXAMPLES:

sage: C = Constellations([[3,1],[3,1],[2,2]]); C
Connected constellations with profile ([3, 1], [3, 1], [2, 2]) on {1, 2, 3, 4}
sage: C.cardinality()
24
sage: C.first()
Constellation of length 3 and degree 4
g0 (1)(2,3,4)
g1 (1,2,3)(4)
g2 (1,2)(3,4)
sage: C.last()
Constellation of length 3 and degree 4
g0 (1,4,3)(2)
g1 (1,4,2)(3)
g2 (1,2)(3,4)

Note that the cardinality can also be computed using characters of the symmetric group (Frobenius formula):

sage: P = Partitions(4)
sage: p1 = Partition([3,1])
sage: p2 = Partition([3,1])
sage: p3 = Partition([2,2])
sage: i1 = P.cardinality() - P.rank(p1) - 1
sage: i2 = P.cardinality() - P.rank(p2) - 1
sage: i3 = P.cardinality() - P.rank(p3) - 1
sage: s = 0
sage: for c in SymmetricGroup(4).irreducible_characters():
....:     v = c.values()
....:     s += v[i1] * v[i2] * v[i3] / v[0]
sage: c1 = p1.conjugacy_class_size()
sage: c2 = p2.conjugacy_class_size()
sage: c3 = p3.conjugacy_class_size()
sage: c1 * c2 * c3 / factorial(4)**2 * s
1

The number obtained above is up to isomorphism. And we can check:

sage: len(C.isomorphism_representatives())
1

isomorphism_representatives()

Return a set of isomorphism representative of self.

EXAMPLES:

sage: C = Constellations([[5], [4,1], [3,2]])
sage: C.cardinality()
240
sage: ir = sorted(C.isomorphism_representatives())
sage: len(ir)
2
sage: ir[0]
Constellation of length 3 and degree 5
g0 (1,2,3,4,5)
g1 (1)(2,3,4,5)
g2 (1,5,3)(2,4)
sage: ir[1]
Constellation of length 3 and degree 5
g0 (1,2,3,4,5)
g1 (1)(2,5,3,4)
g2 (1,5)(2,3,4)

sage.combinat.constellation.perm_conjugate(p, s)

Return the conjugate of the permutation \(p\) by the permutation \(s\).

INPUT:

two permutations of {0,..,n-1} given by lists of values

OUTPUT:

a permutation of {0,..,n-1} given by a list of values

EXAMPLES:

sage: from sage.combinat.constellation import perm_conjugate
sage: perm_conjugate([3,1,2,0], [3,2,0,1])
[0, 3, 2, 1]

sage.combinat.constellation.perm_invert(p)

Return the inverse of the permutation \(p\).

INPUT:

a permutation of {0,..,n-1} given by a list of values

OUTPUT:

a permutation of {0,..,n-1} given by a list of values

EXAMPLES:

sage: from sage.combinat.constellation import perm_invert
sage: perm_invert([3,2,0,1])
[2, 3, 1, 0]

sage.combinat.constellation.perm_sym_domain(g)

Return the domain of a single permutation (before initialization).

EXAMPLES:

sage: from sage.combinat.constellation import perm_sym_domain
sage: perm_sym_domain([1,2,3,4])
{1, 2, 3, 4}
sage: perm_sym_domain(((1,2),(0,4)))
{0, 1, 2, 4}
sage: sorted(perm_sym_domain('(1,2,0,5)'))
[0, 1, 2, 5]

sage.combinat.constellation.perms_are_connected(g, n)

Checks that the action of the generated group is transitive

INPUT:

• a list of permutations of \([0, n-1]\) (in a SymmetricGroup)

• an integer \(n\)

EXAMPLES:

sage: from sage.combinat.constellation import perms_are_connected
sage: S = SymmetricGroup(range(3))
sage: perms_are_connected([S([0,1,2]),S([0,2,1])],3)
False
sage: perms_are_connected([S([0,1,2]),S([1,2,0])],3)
True

sage.combinat.constellation.perms_canonical_labels(p, e=None)

Relabel a list with a common conjugation such that two conjugated lists are relabeled the same way.

INPUT:

• p is a list of at least 2 permutations

• e is None or a list of integer in the domain of the permutations. If provided, then the renumbering algorithm is only performed from the elements of e.

OUTPUT:

• a pair made of a list of permutations (as a list of lists) and a list that corresponds to the conjugacy used.

EXAMPLES:

sage: from sage.combinat.constellation import perms_canonical_labels
sage: l0 = [[2,0,3,1], [3,1,2,0], [0,2,1,3]]
sage: l, m = perms_canonical_labels(l0); l
[[1, 2, 3, 0], [0, 3, 2, 1], [2, 1, 0, 3]]

sage: S = SymmetricGroup(range(4))
sage: [~S(m) * S(u) * S(m) for u in l0] == list(map(S, l))
True

sage: perms_canonical_labels([])
Traceback (most recent call last):
...
ValueError: input must have length >= 2

sage.combinat.constellation.perms_canonical_labels_from(x, y, j0, verbose=False)

Return canonical labels for x, y that starts at j0

Warning

The group generated by x and the elements of y should be transitive.

INPUT:

• x – list - a permutation of \([0, ..., n]\) as a list

• y – list of permutations of \([0, ..., n]\) as a list of lists

• j0 – an index in [0, …, n]

OUTPUT:

mapping: a permutation that specify the new labels

EXAMPLES:

sage: from sage.combinat.constellation import perms_canonical_labels_from
sage: perms_canonical_labels_from([0,1,2],[[1,2,0]], 0)
[0, 1, 2]
sage: perms_canonical_labels_from([1,0,2], [[2,0,1]], 0)
[0, 1, 2]
sage: perms_canonical_labels_from([1,0,2], [[2,0,1]], 1)
[1, 0, 2]
sage: perms_canonical_labels_from([1,0,2], [[2,0,1]], 2)
[2, 1, 0]

sage.combinat.constellation.perms_sym_init(g, sym=None)

Initialize a list of permutations (in the same symmetric group).

OUTPUT:

• sym – a symmetric group

• gg – a list of permutations

EXAMPLES:

sage: from sage.combinat.constellation import perms_sym_init
sage: S, g = perms_sym_init([[0,2,1,3], [1,3,2,0]])
sage: S.domain()
{0, 1, 2, 3}
sage: g
[(1,2), (0,1,3)]

sage: S, g = perms_sym_init(['(2,1)', '(0,3)'])
sage: S.domain()
{0, 1, 2, 3}
sage: g
[(1,2), (0,3)]

sage: S, g = perms_sym_init([(1,0), (2,1)])
sage: S.domain()
{0, 1, 2}
sage: g
[(0,1), (1,2)]

sage: S, g = perms_sym_init([((1,0),(2,3)), '(0,1,4)'])
sage: S.domain()
{0, 1, 2, 3, 4}
sage: g
[(0,1)(2,3), (0,1,4)]