def test_coset_factor():
a = Permutation([0, 2, 1])
G = PermutationGroup([a])
c = Permutation([2, 1, 0])
assert not G.coset_factor(c)
assert G.coset_rank(c) is None
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b])
assert g.order() == 360
d = Permutation([1, 0, 2, 3, 4, 5])
assert not g.coset_factor(d.array_form)
assert not g.contains(d)
assert Permutation(2) in G
c = Permutation([1, 0, 2, 3, 5, 4])
v = g.coset_factor(c, True)
tr = g.basic_transversals
p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
assert p == c
v = g.coset_factor(c)
p = Permutation.rmul(*v)
assert p == c
assert g.contains(c)
G = PermutationGroup([Permutation([2, 1, 0])])
p = Permutation([1, 0, 2])
assert G.coset_factor(p) == []
def test_unrank_lex():
assert Permutation.unrank_lex(5, 10).rank == 10
assert Permutation.unrank_lex(15, 225).rank == 225
assert Permutation.unrank_lex(10, 0).is_Identity
p = Permutation.unrank_lex(4, 23)
assert p.rank == 23
assert p.array_form == [3, 2, 1, 0]
def test_muln():
n = 6
m = 8
a = [Permutation.unrank_nonlex(n, i).array_form for i in range(m)]
h = range(n)
for i in range(m):
h = perm_af_mul(h, a[i])
h2 = perm_af_muln(*a[:i+1])
assert h == h2
def test_mul():
a, b = [0, 2, 1, 3], [0, 1, 3, 2]
assert _af_rmul(a, b) == [0, 2, 3, 1]
assert _af_rmuln(a, b, range(4)) == [0, 2, 3, 1]
assert rmul(Permutation(a), Permutation(b)).array_form == [0, 2, 3, 1]
a = Permutation([0, 2, 1, 3])
b = (0, 1, 3, 2)
c = (3, 1, 2, 0)
assert Permutation.rmul(a, b, c) == Permutation([1, 2, 3, 0])
assert Permutation.rmul(a, c) == Permutation([3, 2, 1, 0])
raises(TypeError, lambda: Permutation.rmul(b, c))
n = 6
m = 8
a = [Permutation.unrank_nonlex(n, i).array_form for i in range(m)]
h = range(n)
for i in range(m):
h = _af_rmul(h, a[i])
h2 = _af_rmuln(*a[:i + 1])
assert h == h2
def test_ranking():
assert Permutation.unrank_lex(5, 10).rank() == 10
p = Permutation.unrank_lex(15, 225)
assert p.rank() == 225
p1 = p.next_lex()
assert p1.rank() == 226
assert Permutation.unrank_lex(15, 225).rank() == 225
assert Permutation.unrank_lex(10, 0).is_Identity
p = Permutation.unrank_lex(4, 23)
assert p.rank() == 23
assert p.array_form == [3, 2, 1, 0]
assert p.next_lex() is None
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
assert a == [[0, 1, 2, 3], [0, 1, 3, 2], [0, 3, 1, 2], [3, 0, 1,
2], [3, 0, 2, 1] ]
assert [Permutation(pa).rank_trotterjohnson() for pa in a] == range(5)
assert Permutation([0, 1, 2, 3]).next_trotterjohnson() == \
Permutation([0, 1, 3, 2])
assert q.rank_trotterjohnson() == 2283
assert p.rank_trotterjohnson() == 3389
assert Permutation([1, 0]).rank_trotterjohnson() == 1
a = Permutation(range(3))
b = a
l = []
tj = []
for i in range(6):
l.append(a)
tj.append(b)
a = a.next_lex()
b = b.next_trotterjohnson()
assert a == b is None
assert set([tuple(a) for a in l]) == set([tuple(a) for a in tj])
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert p.rank() == 1964
assert q.rank() == 870
assert Permutation([]).rank_nonlex() == 0
prank = p.rank_nonlex()
assert prank == 1600
assert Permutation.unrank_nonlex(7, 1600) == p
qrank = q.rank_nonlex()
assert qrank == 41
assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)
a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
assert a == [
[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0],
[2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1],
[1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2],
[2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3],
[3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]
ok = []
p = Permutation([1, 0])
for i in range(3):
ok.append(p.array_form)
p = p.next_nonlex()
if p is None:
ok.append(None)
break
assert ok == [[1, 0], [0, 1], None]
assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)
def test_Permutation():
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[1], [0, 3, 5, 6, 2, 4]])
assert Permutation(p.cyclic_form).array_form == p.array_form
assert p.cardinality == 5040
assert q.cardinality == 5040
assert q.cycles == 2
assert q*p == Permutation([4, 6, 1, 2, 5, 3, 0])
assert p*q == Permutation([6, 5, 3, 0, 2, 4, 1])
assert (Permutation([[1,2,3],[0,4]])*Permutation([[1,2,4],[0],[3]])).cyclic_form == \
[[1, 3], [0, 4, 2]]
assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]
assert p**13 == p
assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
assert q+p == p+q
assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])
a = p-q
b = q-p
assert (a+b).is_Identity
assert len(p.atoms()) == 7
assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])
assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]
assert Permutation.from_inversion_vector(p.inversion_vector()) == p
assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
== q.array_form
assert Permutation([0, 4, 1, 3, 2]).parity() == 0
assert Permutation([0, 1, 4, 3, 2]).parity() == 1
s = Permutation([0])
assert s.is_Singleton
r = Permutation([3, 2, 1, 0])
assert (r**2).is_Identity
assert (p*(~p)).is_Identity
assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
assert ~(r**2).is_Identity
assert p.max() == 6
assert p.min() == 0
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert q.max() == 4
assert q.min() == 0
assert Permutation([]).rank_nonlex() == 0
prank = p.rank_nonlex()
assert prank == 1600
assert Permutation.unrank_nonlex(7, 1600) == p
qrank = q.rank_nonlex()
assert qrank == 41
assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)
a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
assert a == \
[[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0], \
[2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1], \
[1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2], \
[2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3], \
[3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]
assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)
assert q.rank() == 870
assert p.rank() == 1964
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
assert p.ascents() == [0, 3, 4]
assert q.ascents() == [1, 2, 4]
assert r.ascents() == []
assert p.descents() == [1, 2, 5]
assert q.descents() == [0, 3, 5]
assert Permutation(r.descents()).is_Identity
assert p.inversions() == 7
assert p.signature() == -1
assert q.inversions() == 11
assert q.signature() == -1
assert (p*(~p)).inversions() == 0
assert (p*(~p)).signature() == 1
assert p.order() == 6
assert q.order() == 10
assert (p**(p.order())).is_Identity
assert p.length() == 6
assert q.length() == 7
assert r.length() == 4
assert not p.is_Positive()
assert p.is_Negative()
assert not q.is_Positive()
assert q.is_Negative()
assert r.is_Positive()
assert not r.is_Negative()
assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
assert r.runs() == [[3], [2], [1], [0]]
assert p.index() == 8
assert q.index() == 8
assert r.index() == 3
a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
assert a == [[0,1,2,3], [0,1,3,2], [0,3,1,2], [3,0,1,2], [3,0,2,1] ]
assert [Permutation(pa).rank_trotterjohnson() for pa in a] == range(5)
assert q.rank_trotterjohnson() == 2283
assert p.rank_trotterjohnson() == 3389
assert p.get_precedence_distance(q) == q.get_precedence_distance(p)
assert p.get_adjacency_distance(q) == p.get_adjacency_distance(q)
assert p.get_positional_distance(q) == p.get_positional_distance(q)
p = Permutation([0, 1, 2, 3])
q = Permutation([3, 2, 1, 0])
assert p.get_precedence_distance(q) == 6
assert p.get_adjacency_distance(q) == 3
assert p.get_positional_distance(q) == 8
def test_polyhedron():
raises(ValueError,
lambda: Polyhedron(list('ab'), pgroup=[Permutation([0])]))
pgroup = [Permutation([[0,7,2,5],[6,1,4,3]]),\
Permutation([[0,7,1,6],[5,2,4,3]]),\
Permutation([[3,6,0,5],[4,1,7,2]]),\
Permutation([[7,4,5],[1,3,0],[2],[6]]),\
Permutation([[1,3,2],[7,6,5],[4],[0]]),\
Permutation([[4,7,6],[2,0,3],[1],[5]]),\
Permutation([[1,2,0],[4,5,6],[3],[7]]),\
Permutation([[4,2],[0,6],[3,7],[1,5]]),\
Permutation([[3,5],[7,1],[2,6],[0,4]]),\
Permutation([[2,5],[1,6],[0,4],[3,7]]),\
Permutation([[4,3],[7,0],[5,1],[6,2]]),\
Permutation([[4,1],[0,5],[6,2],[7,3]]),\
Permutation([[7,2],[3,6],[0,4],[1,5]]),\
Permutation([0,1,2,3,4,5,6,7])]
corners = tuple(symbols('A:H'))
faces = cube_faces
cube = Polyhedron(corners, faces, pgroup)
assert cube.edges == FiniteSet(*((0, 1), (6, 7), (1, 2), (5, 6), (0, 3),
(2, 3), (4, 7), (4, 5), (3, 7), (1, 5),
(0, 4), (2, 6)))
for i in xrange(3): # add 180 degree face rotations
cube.rotate(cube.pgroup[i]**2)
assert cube.corners == corners
for i in range(3, 7): # add 240 degree axial corner rotations
cube.rotate(cube.pgroup[i]**2)
assert cube.corners == corners
cube.rotate(1)
raises(ValueError, lambda: cube.rotate(Permutation([0, 1])))
assert cube.corners != corners
assert cube.array_form == [7, 6, 4, 5, 3, 2, 0, 1]
assert cube.cyclic_form == [[0, 7, 1, 6], [2, 4, 3, 5]]
cube.reset()
assert cube.corners == corners
def check(h, size, rpt, target):
assert len(h.faces) + len(h.vertices) - len(h.edges) == 2
assert h.size == size
got = set()
for p in h.pgroup:
# make sure it restores original
P = h.copy()
hit = P.corners
for i in range(rpt):
P.rotate(p)
if P.corners == hit:
break
else:
print 'error in permutation', p.array_form
for i in range(rpt):
P.rotate(p)
got.add(tuple(P.corners))
c = P.corners
f = [[c[i] for i in f] for f in P.faces]
assert h.faces == Polyhedron(c, f).faces
assert len(got) == target
assert PermutationGroup([Permutation(g) for g in got]).is_group()
for h, size, rpt, target in zip(
(tetrahedron, square, octahedron, dodecahedron, icosahedron),
(4, 8, 6, 20, 12), (3, 4, 4, 5, 5), (12, 24, 24, 60, 60)):
check(h, size, rpt, target)
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
S = SymmetricGroup(10)
N_eps = 10
_random_prec = {
'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])
}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
_random_prec = {
'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])
}
assert A.is_alt_sym(_random_prec=_random_prec) is False
def test_is_trivial():
for i in range(5):
triv = PermutationGroup([Permutation(list(range(i)))])
assert triv.is_trivial
def test_orbit_rep():
G = DihedralGroup(6)
assert G.orbit_rep(1,3) in [Permutation([2, 3, 4, 5, 0, 1]),\
Permutation([4, 3, 2, 1, 0, 5])]
H = CyclicGroup(4)*G
assert H.orbit_rep(1, 5) == False
def test_coset_factor():
a = Permutation([0, 2, 1])
G = PermutationGroup([a])
c = Permutation([2, 1, 0])
assert not G.coset_factor(c)
assert G.coset_rank(c) is None
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b])
assert g.order() == 360
d = Permutation([1, 0, 2, 3, 4, 5])
assert not g.coset_factor(d.array_form)
assert not g.contains(d)
assert Permutation(2) in G
c = Permutation([1, 0, 2, 3, 5, 4])
v = g.coset_factor(c, True)
tr = g.basic_transversals
p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
assert p == c
v = g.coset_factor(c)
p = Permutation.rmul(*v)
assert p == c
assert g.contains(c)
G = PermutationGroup([Permutation([2, 1, 0])])
p = Permutation([1, 0, 2])
assert G.coset_factor(p) == []
def SymmetricGroup(n):
"""
Generates the symmetric group on ``n`` elements as a permutation group.
The generators taken are the ``n``-cycle
``(0 1 2 ... n-1)`` and the transposition ``(0 1)`` (in cycle notation).
(See [1]). After the group is generated, some of its basic properties
are set.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(4)
>>> G.is_group
True
>>> G.order()
24
>>> list(G.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1],
[1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3],
[2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0],
[3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0],
[0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]]
See Also
========
CyclicGroup, DihedralGroup, AlternatingGroup
References
==========
.. [1] https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations
"""
if n == 1:
G = PermutationGroup([Permutation([0])])
elif n == 2:
G = PermutationGroup([Permutation([1, 0])])
else:
a = list(range(1, n))
a.append(0)
gen1 = _af_new(a)
a = list(range(n))
a[0], a[1] = a[1], a[0]
gen2 = _af_new(a)
G = PermutationGroup([gen1, gen2])
if n < 3:
G._is_abelian = True
G._is_nilpotent = True
else:
G._is_abelian = False
G._is_nilpotent = False
if n < 5:
G._is_solvable = True
else:
G._is_solvable = False
G._degree = n
G._is_transitive = True
G._is_sym = True
return G
def AlternatingGroup(n):
"""
Generates the alternating group on ``n`` elements as a permutation group.
For ``n > 2``, the generators taken are ``(0 1 2), (0 1 2 ... n-1)`` for
``n`` odd
and ``(0 1 2), (1 2 ... n-1)`` for ``n`` even (See [1], p.31, ex.6.9.).
After the group is generated, some of its basic properties are set.
The cases ``n = 1, 2`` are handled separately.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> G = AlternatingGroup(4)
>>> G.is_group
True
>>> a = list(G.generate_dimino())
>>> len(a)
12
>>> all(perm.is_even for perm in a)
True
See Also
========
SymmetricGroup, CyclicGroup, DihedralGroup
References
==========
[1] Armstrong, M. "Groups and Symmetry"
"""
# small cases are special
if n in (1, 2):
return PermutationGroup([Permutation([0])])
a = list(range(n))
a[0], a[1], a[2] = a[1], a[2], a[0]
gen1 = a
if n % 2:
a = list(range(1, n))
a.append(0)
gen2 = a
else:
a = list(range(2, n))
a.append(1)
a.insert(0, 0)
gen2 = a
gens = [gen1, gen2]
if gen1 == gen2:
gens = gens[:1]
G = PermutationGroup([_af_new(a) for a in gens], dups=False)
if n < 4:
G._is_abelian = True
G._is_nilpotent = True
else:
G._is_abelian = False
G._is_nilpotent = False
if n < 5:
G._is_solvable = True
else:
G._is_solvable = False
G._degree = n
G._is_transitive = True
G._is_alt = True
return G
def test_PermutationGroup():
assert PermutationGroup() == PermutationGroup(Permutation())
assert (PermutationGroup() == 0) is False
def DihedralGroup(n):
r"""
Generates the dihedral group `D_n` as a permutation group.
The dihedral group `D_n` is the group of symmetries of the regular
``n``-gon. The generators taken are the ``n``-cycle ``a = (0 1 2 ... n-1)``
(a rotation of the ``n``-gon) and ``b = (0 n-1)(1 n-2)...``
(a reflection of the ``n``-gon) in cycle rotation. It is easy to see that
these satisfy ``a**n = b**2 = 1`` and ``bab = ~a`` so they indeed generate
`D_n` (See [1]). After the group is generated, some of its basic properties
are set.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(5)
>>> G.is_group
True
>>> a = list(G.generate_dimino())
>>> [perm.cyclic_form for perm in a]
[[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],
[[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]],
[[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]],
[[0, 3], [1, 2]]]
See Also
========
SymmetricGroup, CyclicGroup, AlternatingGroup
References
==========
[1] https://en.wikipedia.org/wiki/Dihedral_group
"""
# small cases are special
if n == 1:
return PermutationGroup([Permutation([1, 0])])
if n == 2:
return PermutationGroup([Permutation([1, 0, 3, 2]),
Permutation([2, 3, 0, 1]), Permutation([3, 2, 1, 0])])
a = list(range(1, n))
a.append(0)
gen1 = _af_new(a)
a = list(range(n))
a.reverse()
gen2 = _af_new(a)
G = PermutationGroup([gen1, gen2])
# if n is a power of 2, group is nilpotent
if n & (n-1) == 0:
G._is_nilpotent = True
else:
G._is_nilpotent = False
G._is_abelian = False
G._is_solvable = True
G._degree = n
G._is_transitive = True
G._order = 2*n
return G
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
assert G._eval_is_alt_sym_naive() is False
assert G._eval_is_alt_sym_naive(only_alt=True) is False
assert G._eval_is_alt_sym_naive(only_sym=True) is False
S = SymmetricGroup(10)
assert S._eval_is_alt_sym_naive() is True
assert S._eval_is_alt_sym_naive(only_alt=True) is False
assert S._eval_is_alt_sym_naive(only_sym=True) is True
N_eps = 10
_random_prec = {
'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])
}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
assert A._eval_is_alt_sym_naive() is True
assert A._eval_is_alt_sym_naive(only_alt=True) is True
assert A._eval_is_alt_sym_naive(only_sym=True) is False
_random_prec = {
'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])
}
assert A.is_alt_sym(_random_prec=_random_prec) is False
G = PermutationGroup(
Permutation(1, 3, size=8)(0, 2, 4, 6),
Permutation(5, 7, size=8)(0, 2, 4, 6))
assert G.is_alt_sym() is False
# Tests for monte-carlo c_n parameter setting, and which guarantees
# to give False.
G = DihedralGroup(10)
assert G._eval_is_alt_sym_monte_carlo() is False
G = DihedralGroup(20)
assert G._eval_is_alt_sym_monte_carlo() is False
# A dry-running test to check if it looks up for the updated cache.
G = DihedralGroup(6)
G.is_alt_sym()
assert G.is_alt_sym() == False
def test_cyclic():
G = SymmetricGroup(2)
assert G.is_cyclic
G = AbelianGroup(3, 7)
assert G.is_cyclic
G = AbelianGroup(7, 7)
assert not G.is_cyclic
G = AlternatingGroup(3)
assert G.is_cyclic
G = AlternatingGroup(4)
assert not G.is_cyclic
# Order less than 6
G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 1))
assert G.is_cyclic
G = PermutationGroup(Permutation(0, 1, 2, 3), Permutation(0, 2)(1, 3))
assert G.is_cyclic
G = PermutationGroup(Permutation(3),
Permutation(0, 1)(2, 3),
Permutation(0, 2)(1, 3),
Permutation(0, 3)(1, 2))
assert G.is_cyclic is False
# Order 15
G = PermutationGroup(
Permutation(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14),
Permutation(0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13))
assert G.is_cyclic
# Distinct prime orders
assert PermutationGroup._distinct_primes_lemma([3, 5]) is True
assert PermutationGroup._distinct_primes_lemma([5, 7]) is True
assert PermutationGroup._distinct_primes_lemma([2, 3]) is None
assert PermutationGroup._distinct_primes_lemma([3, 5, 7]) is None
assert PermutationGroup._distinct_primes_lemma([5, 7, 13]) is True
G = PermutationGroup(Permutation(0, 1, 2, 3), Permutation(0, 2)(1, 3))
assert G.is_cyclic
assert G._is_abelian
def test_index():
G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 3))
H = G.subgroup([Permutation(0, 1, 3)])
assert G.index(H) == 4
def test_from_sequence():
assert Permutation.from_sequence('SymPy') == Permutation(4)(0, 1, 3)
assert Permutation.from_sequence('SymPy', key=lambda x: x.lower()) == \
Permutation(4)(0, 2)(1, 3)
def test_orbits():
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
g = PermutationGroup([a, b])
assert g.orbit(0) == {0, 1, 2}
assert g.orbits() == [{0, 1, 2}]
assert g.is_transitive() and g.is_transitive(strict=False)
assert g.orbit_transversal(0) == \
[Permutation(
[0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
assert g.orbit_transversal(0, True) == \
[(0, Permutation([0, 1, 2])), (2, Permutation([2, 0, 1])),
(1, Permutation([1, 2, 0]))]
a = Permutation(list(range(1, 100)) + [0])
G = PermutationGroup([a])
assert [min(o) for o in G.orbits()] == [0]
G = PermutationGroup(rubik_cube_generators())
assert [min(o) for o in G.orbits()] == [0, 1]
assert not G.is_transitive() and not G.is_transitive(strict=False)
G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
assert not G.is_transitive() and G.is_transitive(strict=False)
assert PermutationGroup(
Permutation(3)).is_transitive(strict=False) is False
def test_ranking():
assert Permutation.unrank_lex(5, 10).rank() == 10
p = Permutation.unrank_lex(15, 225)
assert p.rank() == 225
p1 = p.next_lex()
assert p1.rank() == 226
assert Permutation.unrank_lex(15, 225).rank() == 225
assert Permutation.unrank_lex(10, 0).is_Identity
p = Permutation.unrank_lex(4, 23)
assert p.rank() == 23
assert p.array_form == [3, 2, 1, 0]
assert p.next_lex() is None
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
assert a == [[0, 1, 2, 3], [0, 1, 3, 2], [0, 3, 1, 2], [3, 0, 1, 2],
[3, 0, 2, 1]]
assert [Permutation(pa).rank_trotterjohnson()
for pa in a] == list(range(5))
assert Permutation([0, 1, 2, 3]).next_trotterjohnson() == \
Permutation([0, 1, 3, 2])
assert q.rank_trotterjohnson() == 2283
assert p.rank_trotterjohnson() == 3389
assert Permutation([1, 0]).rank_trotterjohnson() == 1
a = Permutation(list(range(3)))
b = a
l = []
tj = []
for i in range(6):
l.append(a)
tj.append(b)
a = a.next_lex()
b = b.next_trotterjohnson()
assert a == b is None
assert {tuple(a) for a in l} == {tuple(a) for a in tj}
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert p.rank() == 1964
assert q.rank() == 870
assert Permutation([]).rank_nonlex() == 0
prank = p.rank_nonlex()
assert prank == 1600
assert Permutation.unrank_nonlex(7, 1600) == p
qrank = q.rank_nonlex()
assert qrank == 41
assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)
a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
assert a == [[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3],
[2, 3, 1, 0], [2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3],
[1, 3, 2, 0], [3, 0, 2, 1], [1, 0, 3, 2], [1, 0, 2, 3],
[2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2], [2, 1, 0, 3],
[3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3],
[3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]
N = 10
p1 = Permutation(a[0])
for i in range(1, N + 1):
p1 = p1 * Permutation(a[i])
p2 = Permutation.rmul_with_af(*[Permutation(h) for h in a[N::-1]])
assert p1 == p2
ok = []
p = Permutation([1, 0])
for i in range(3):
ok.append(p.array_form)
p = p.next_nonlex()
if p is None:
ok.append(None)
break
assert ok == [[1, 0], [0, 1], None]
assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
assert [Permutation(pa).rank_nonlex() for pa in a] == list(range(24))
def test_elements():
p = Permutation(2, 3)
assert PermutationGroup(p).elements == {Permutation(3), Permutation(2, 3)}
def test_coset_repr():
a = Permutation([0, 2, 1])
b = Permutation([1, 0, 2])
G = PermutationGroup([a, b])
assert G.coset_repr() == [[[0,1,2], [1,0,2], [2,0,1]], [[0,1,2], [0,2,1]]]
assert G.stabilizers_gens() == [[0, 2, 1]]
def test_is_group():
assert PermutationGroup(Permutation(1,2), Permutation(2,4)).is_group == True
assert SymmetricGroup(4).is_group == True
def test_orbits():
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
g = PermutationGroup([a, b])
assert g.orbit(0) == {0, 1, 2}
assert g.orbits() == [{0, 1, 2}]
assert g.is_transitive() and g.is_transitive(strict=False)
assert g.orbit_transversal(0) == \
[Permutation(
[0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
assert g.orbit_transversal(0, True) == \
[(0, Permutation([0, 1, 2])), (2, Permutation([2, 0, 1])),
(1, Permutation([1, 2, 0]))]
G = DihedralGroup(6)
transversal, slps = _orbit_transversal(G.degree,
G.generators,
0,
True,
slp=True)
for i, t in transversal:
slp = slps[i]
w = G.identity
for s in slp:
w = G.generators[s] * w
assert w == t
a = Permutation(list(range(1, 100)) + [0])
G = PermutationGroup([a])
assert [min(o) for o in G.orbits()] == [0]
G = PermutationGroup(rubik_cube_generators())
assert [min(o) for o in G.orbits()] == [0, 1]
assert not G.is_transitive() and not G.is_transitive(strict=False)
G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
assert not G.is_transitive() and G.is_transitive(strict=False)
assert PermutationGroup(
Permutation(3)).is_transitive(strict=False) is False
def test_Permutation():
p = Permutation([2,5,1,6,3,0,4])
q = Permutation([[1,4,5],[2,0,6],[3]])
assert q.cycles == 3
assert p*q == Permutation([4, 6, 1, 2, 5, 3, 0])
assert q*p == Permutation([6, 5, 3, 0, 2, 4, 1])
assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]
assert p**13 == p
assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
assert q+p == p+q
assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])
a = p-q
b = q-p
assert (a+b).is_Identity
assert len(p.atoms()) == 7
assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])
assert p.inversion_vector == [2, 4, 1, 3, 1, 0]
assert q.inversion_vector == [3, 1, 2, 2, 0, 1]
assert Permutation.from_inversion_vector(p.inversion_vector) == p
assert Permutation.from_inversion_vector(q.inversion_vector).array_form\
== q.array_form
s = Permutation([0])
assert s.is_Singleton
r = Permutation([3,2,1,0])
assert (r**2).is_Identity
assert (p*(~p)).is_Identity
assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
assert ~(r**2).is_Identity
assert p.max == 6
assert p.min == 0
q = Permutation([[4,1,2,3],[0,5,6]])
assert q.max == 4
assert q.min == 0
assert p.rank_nonlex() == 14830
assert q.rank_nonlex() == 8441
assert Permutation.unrank_nonlex(7, 41) == Permutation([4, 2, 3, 5, 1, 0, 6])
assert q.rank == 870
assert p.rank == 1964
p = Permutation([1,5,2,0,3,6,4])
q = Permutation([[2,3,5],[1,0,6],[4]])
assert p.ascents == [0, 3, 4]
assert q.ascents == [1, 2, 4]
assert r.ascents == []
assert p.descents == [1, 2, 5]
assert q.descents == [0, 3, 5]
assert Permutation(r.descents).is_Identity
assert p.inversions == 7
assert p.signature == -1
assert q.inversions == 11
assert q.signature == -1
assert (p*(~p)).inversions == 0
assert (p*(~p)).signature == 1
assert p.order == 6
assert q.order == 3
assert (p**(p.order)).is_Identity
assert p.length == 6
assert q.length == 7
assert r.length == 4
assert not p.is_Positive
assert p.is_Negative
assert not q.is_Positive
assert q.is_Negative
assert r.is_Positive
assert not r.is_Negative
assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
assert r.runs() == [[3], [2], [1], [0]]
assert p.index == 8
assert q.index == 8
assert r.index == 3
assert q.rank_trotterjohnson == 259
assert p.rank_trotterjohnson == 1087
assert p.get_precedence_distance(q) == q.get_precedence_distance(p)
assert p.get_adjacency_distance(q) == p.get_adjacency_distance(q)
assert p.get_positional_distance(q) == p.get_positional_distance(q)
p = Permutation([0, 1, 2, 3])
q = Permutation([3, 2, 1, 0])
assert p.get_precedence_distance(q) == 6
assert p.get_adjacency_distance(q) == 3
assert p.get_positional_distance(q) == 8
def test_order():
a = Permutation([2, 0, 1, 3, 4, 5, 6, 7, 8, 9])
b = Permutation([2, 1, 3, 4, 5, 6, 7, 8, 9, 0])
g = PermutationGroup([a, b])
assert g.order() == 1814400
assert PermutationGroup().order() == 1
def test_from_sequence():
assert Permutation.from_sequence('SymPy') == Permutation(4)(0, 1, 3)
assert Permutation.from_sequence('SymPy', key=lambda x: x.lower()) == \
Permutation(4)(0, 2)(1, 3)
def test_make_perm():
assert cube.pgroup.make_perm(5, seed=list(range(5))) == \
Permutation([4, 7, 6, 5, 0, 3, 2, 1])
assert cube.pgroup.make_perm(7, seed=list(range(7))) == \
Permutation([6, 7, 3, 2, 5, 4, 0, 1])
def test_args():
p = Permutation([(0, 3, 1, 2), (4, 5)])
assert p._cyclic_form is None
assert Permutation(p) == p
assert p.cyclic_form == [[0, 3, 1, 2], [4, 5]]
assert p._array_form == [3, 2, 0, 1, 5, 4]
p = Permutation((0, 3, 1, 2))
assert p._cyclic_form is None
assert p._array_form == [0, 3, 1, 2]
assert Permutation([0]) == Permutation((0, ))
assert Permutation([[0], [1]]) == Permutation(((0, ), (1, ))) == \
Permutation(((0, ), [1]))
assert Permutation([[1, 2]]) == Permutation([0, 2, 1])
assert Permutation([[1], [4, 2]]) == Permutation([0, 1, 4, 3, 2])
assert Permutation([[1], [4, 2]], size=1) == Permutation([0, 1, 4, 3, 2])
assert Permutation([[1], [4, 2]],
size=6) == Permutation([0, 1, 4, 3, 2, 5])
assert Permutation([], size=3) == Permutation([0, 1, 2])
assert Permutation(3).list(5) == [0, 1, 2, 3, 4]
assert Permutation(3).list(-1) == []
assert Permutation(5)(1, 2).list(-1) == [0, 2, 1]
assert Permutation(5)(1, 2).list() == [0, 2, 1, 3, 4, 5]
raises(ValueError, lambda: Permutation([1, 2], [0]))
# enclosing brackets needed
raises(ValueError, lambda: Permutation([[1, 2], 0]))
# enclosing brackets needed on 0
raises(ValueError, lambda: Permutation([1, 1, 0]))
raises(ValueError, lambda: Permutation([[1], [1, 2]]))
raises(ValueError, lambda: Permutation([4, 5], size=10)) # where are 0-3?
# but this is ok because cycles imply that only those listed moved
assert Permutation(4, 5) == Permutation([0, 1, 2, 3, 5, 4])
def test_PermutationGroup():
assert PermutationGroup() == PermutationGroup(Permutation())
def test_sympy__combinatorics__permutations__Permutation():
from sympy.combinatorics.permutations import Permutation
assert _test_args(Permutation([0, 1, 2, 3]))
def test_ranking():
assert Permutation.unrank_lex(5, 10).rank() == 10
p = Permutation.unrank_lex(15, 225)
assert p.rank() == 225
p1 = p.next_lex()
assert p1.rank() == 226
assert Permutation.unrank_lex(15, 225).rank() == 225
assert Permutation.unrank_lex(10, 0).is_Identity
p = Permutation.unrank_lex(4, 23)
assert p.rank() == 23
assert p.array_form == [3, 2, 1, 0]
assert p.next_lex() == None
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
assert a == [[0,1,2,3], [0,1,3,2], [0,3,1,2], [3,0,1,2], [3,0,2,1] ]
assert [Permutation(pa).rank_trotterjohnson() for pa in a] == range(5)
assert Permutation([0,1,2,3]).next_trotterjohnson() == \
Permutation([0,1,3,2])
assert q.rank_trotterjohnson() == 2283
assert p.rank_trotterjohnson() == 3389
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert p.rank() == 1964
assert q.rank() == 870
assert Permutation([]).rank_nonlex() == 0
prank = p.rank_nonlex()
assert prank == 1600
assert Permutation.unrank_nonlex(7, 1600) == p
qrank = q.rank_nonlex()
assert qrank == 41
assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)
a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
assert a == \
[[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0], \
[2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1], \
[1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2], \
[2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3], \
[3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]
assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)
def test_coset_transvesal():
G = AlternatingGroup(5)
H = PermutationGroup(Permutation(0,1,2),Permutation(1,2)(3,4))
assert G.coset_transversal(H) == \
[Permutation(4), Permutation(2, 3, 4), Permutation(2, 4, 3),
Permutation(1, 2, 4), Permutation(4)(1, 2, 3), Permutation(1, 3)(2, 4),
Permutation(0, 1, 2, 3, 4), Permutation(0, 1, 2, 4, 3),
Permutation(0, 1, 3, 2, 4), Permutation(0, 2, 4, 1, 3)]
def test_Permutation():
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[1], [0, 3, 5, 6, 2, 4]])
assert Permutation(p.cyclic_form).array_form == p.array_form
assert p.cardinality == 5040
assert q.cardinality == 5040
assert q.cycles == 2
assert q*p == Permutation([4, 6, 1, 2, 5, 3, 0])
assert p*q == Permutation([6, 5, 3, 0, 2, 4, 1])
assert perm_af_mul([2, 5, 1, 6, 3, 0, 4], [3, 1, 4, 5, 0, 6, 2]) == \
[6, 5, 3, 0, 2, 4, 1]
assert cyclic([(2,3,5)], 5) == [[1, 2, 4], [0], [3]]
assert (Permutation([[1,2,3],[0,4]])*Permutation([[1,2,4],[0],[3]])).cyclic_form == \
[[1, 3], [0, 4, 2]]
assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]
assert p.transpositions() == [(3, 4), (3, 6), (0, 5), (0, 1), (0, 2)]
assert p**13 == p
assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
assert q+p == p+q
assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])
a = p-q
b = q-p
assert (a+b).is_Identity
assert p.conjugate(q) == Permutation([5, 3, 0, 4, 6, 2, 1])
assert p.conjugate(q) == ~q*p*q == p**q
assert q.conjugate(p) == Permutation([6, 3, 2, 0, 1, 4, 5])
assert q.conjugate(p) == ~p*q*p == q**p
assert p.commutator(q) == Permutation([1, 4, 5, 6, 3, 0, 2])
assert q.commutator(p) == Permutation([5, 0, 6, 4, 1, 2, 3])
assert p.commutator(q) == ~ q.commutator(p)
assert len(p.atoms()) == 7
assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])
assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]
assert Permutation.from_inversion_vector(p.inversion_vector()) == p
assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
== q.array_form
assert Permutation([i for i in range(500,-1,-1)]).inversions() == 125250
assert Permutation([0, 4, 1, 3, 2]).parity() == 0
assert Permutation([0, 1, 4, 3, 2]).parity() == 1
assert perm_af_parity([0, 4, 1, 3, 2]) == 0
assert perm_af_parity([0, 1, 4, 3, 2]) == 1
s = Permutation([0])
assert s.is_Singleton
r = Permutation([3, 2, 1, 0])
assert (r**2).is_Identity
assert (p*(~p)).is_Identity
assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
assert ~(r**2).is_Identity
assert p.max() == 6
assert p.min() == 0
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert q.max() == 4
assert q.min() == 0
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
assert p.ascents() == [0, 3, 4]
assert q.ascents() == [1, 2, 4]
assert r.ascents() == []
assert p.descents() == [1, 2, 5]
assert q.descents() == [0, 3, 5]
assert Permutation(r.descents()).is_Identity
assert p.inversions() == 7
assert p.signature() == -1
assert q.inversions() == 11
assert q.signature() == -1
assert (p*(~p)).inversions() == 0
assert (p*(~p)).signature() == 1
assert p.order() == 6
assert q.order() == 10
assert (p**(p.order())).is_Identity
assert p.length() == 6
assert q.length() == 7
assert r.length() == 4
assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
assert r.runs() == [[3], [2], [1], [0]]
assert p.index() == 8
assert q.index() == 8
assert r.index() == 3
assert p.get_precedence_distance(q) == q.get_precedence_distance(p)
assert p.get_adjacency_distance(q) == p.get_adjacency_distance(q)
assert p.get_positional_distance(q) == p.get_positional_distance(q)
p = Permutation([0, 1, 2, 3])
q = Permutation([3, 2, 1, 0])
assert p.get_precedence_distance(q) == 6
assert p.get_adjacency_distance(q) == 3
assert p.get_positional_distance(q) == 8
a = [Permutation.unrank_nonlex(4, i) for i in range(5)]
iden = Permutation([0, 1, 2, 3])
for i in range(5):
for j in range(i+1, 5):
assert a[i].commutes_with(a[j]) == (a[i]*a[j] == a[j]*a[i])
if a[i].commutes_with(a[j]):
assert a[i].commutator(a[j]) == iden
assert a[j].commutator(a[i]) == iden
def test_subgroup():
G = PermutationGroup(Permutation(0,1,2), Permutation(0,2,3))
H = G.subgroup([Permutation(0,1,3)])
assert H.is_subgroup(G)
def test_josephus():
assert Permutation.josephus(4, 6, 1) == Permutation([3, 1, 0, 2, 5, 4])
assert Permutation.josephus(1, 5, 1).is_Identity
def test_nC_nP_nT():
from sympy.utilities.iterables import (multiset_permutations,
multiset_combinations,
multiset_partitions, partitions,
subsets, permutations)
from sympy.functions.combinatorial.numbers import (nP, nC, nT, stirling,
_stirling1, _stirling2,
_multiset_histogram,
_AOP_product)
from sympy.combinatorics.permutations import Permutation
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
assert check.is_Integer
tot += check
assert sum(1 for p in partitions(i, j, size=True)
if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(list(range(i)), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)
] == [0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [
sum(1 for p in perms if Permutation(p).cycles == i) for i in range(5)
] == [0, 6, 11, 6, 1] == [stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [
stirling(n, k, signed=1) for n in range(10) for k in range(1, n + 1)
] == [
1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50, 35, -10, 1, -120, 274,
-225, 85, -15, 1, 720, -1764, 1624, -735, 175, -21, 1, -5040, 13068,
-13132, 6769, -1960, 322, -28, 1, 40320, -109584, 118124, -67284,
22449, -4536, 546, -36, 1
]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1) for n in range(10)
for k in range(n + 1)] == [
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35,
10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735,
175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0,
40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1
]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2) for n in range(10)
for k in range(n + 1)] == [
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10,
1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0,
1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770,
6951, 2646, 462, 36, 1
]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
# Assertion that the return type is SymPy Integer.
assert isinstance(_stirling1(6, 3), Integer)
assert isinstance(_stirling2(6, 3), Integer)
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1
for p in parts if all(delta(i) >= d
for i in p)) == stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb' * 3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4, 1, 1, 1))) == {
0: 1,
1: 4,
2: 7,
3: 8,
4: 8,
5: 7,
6: 4,
7: 1
}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1: 'a'}))
def mult(permutation_a, permutation_b):
return Permutation(
[permutation_a[j] for i, j in enumerate(permutation_b.array_forms)])
def test1():
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([0, 2, 1, 3, 4])
g = PermutationGroup([a, b])
def test_Permutation():
# don't auto fill 0
raises(ValueError, lambda: Permutation([1]))
p = Permutation([0, 1, 2, 3])
# call as bijective
assert [p(i) for i in range(p.size)] == list(p)
# call as operator
assert p(range(p.size)) == list(p)
# call as function
assert list(p(1, 2)) == [0, 2, 1, 3]
# conversion to list
assert list(p) == range(4)
# cycle form with size
assert Permutation([[1, 2]], size=4) == Permutation([[1, 2], [0], [3]])
# random generation
assert Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1]))
p = Permutation([2, 5, 1, 6, 3, 0, 4])
q = Permutation([[1], [0, 3, 5, 6, 2, 4]])
assert len(set([p, p])) == 1
r = Permutation([1, 3, 2, 0, 4, 6, 5])
ans = Permutation(_af_rmuln(*[w.array_form for w in (p, q, r)])).array_form
assert rmul(p, q, r).array_form == ans
# make sure no other permutation of p, q, r could have given
# that answer
for a, b, c in permutations((p, q, r)):
if (a, b, c) == (p, q, r):
continue
assert rmul(a, b, c).array_form != ans
assert p.support() == range(7)
assert q.support() == [0, 2, 3, 4, 5, 6]
assert Permutation(p.cyclic_form).array_form == p.array_form
assert p.cardinality == 5040
assert q.cardinality == 5040
assert q.cycles == 2
assert rmul(q, p) == Permutation([4, 6, 1, 2, 5, 3, 0])
assert rmul(p, q) == Permutation([6, 5, 3, 0, 2, 4, 1])
assert _af_rmul(p.array_form, q.array_form) == \
[6, 5, 3, 0, 2, 4, 1]
assert rmul(Permutation([[1, 2, 3], [0, 4]]),
Permutation([[1, 2, 4], [0], [3]])).cyclic_form == \
[[0, 4, 2], [1, 3]]
assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
assert q.cyclic_form == [[0, 3, 5, 6, 2, 4]]
assert q.full_cyclic_form == [[0, 3, 5, 6, 2, 4], [1]]
assert p.cyclic_form == [[0, 2, 1, 5], [3, 6, 4]]
t = p.transpositions()
assert t == [(0, 5), (0, 1), (0, 2), (3, 4), (3, 6)]
assert Permutation.rmul(*[Permutation(Cycle(*ti)) for ti in (t)])
assert Permutation([1, 0]).transpositions() == [(0, 1)]
assert p**13 == p
assert q**0 == Permutation(range(q.size))
assert q**-2 == ~q**2
assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
assert q**3 == q**2*q
assert q**4 == q**2*q**2
a = Permutation(1, 3)
b = Permutation(2, 0, 3)
I = Permutation(3)
assert ~a == a**-1
assert a*~a == I
assert a*b**-1 == a*~b
ans = Permutation(0, 5, 3, 1, 6)(2, 4)
assert (p + q.rank()).rank() == ans.rank()
assert (p + q.rank())._rank == ans.rank()
assert (q + p.rank()).rank() == ans.rank()
raises(TypeError, lambda: p + Permutation(range(10)))
assert (p - q.rank()).rank() == Permutation(0, 6, 3, 1, 2, 5, 4).rank()
assert p.rank() - q.rank() < 0 # for coverage: make sure mod is used
assert (q - p.rank()).rank() == Permutation(1, 4, 6, 2)(3, 5).rank()
assert p*q == Permutation(_af_rmuln(*[list(w) for w in (q, p)]))
assert p*Permutation([]) == p
assert Permutation([])*p == p
assert p*Permutation([[0, 1]]) == Permutation([2, 5, 0, 6, 3, 1, 4])
assert Permutation([[0, 1]])*p == Permutation([5, 2, 1, 6, 3, 0, 4])
pq = p^q
assert pq == Permutation([5, 6, 0, 4, 1, 2, 3])
assert pq == rmul(q, p, ~q)
qp = q^p
assert qp == Permutation([4, 3, 6, 2, 1, 5, 0])
assert qp == rmul(p, q, ~p)
raises(ValueError, lambda: p^Permutation([]))
assert p.commutator(q) == Permutation(0, 1, 3, 4, 6, 5, 2)
assert q.commutator(p) == Permutation(0, 2, 5, 6, 4, 3, 1)
assert p.commutator(q) == ~q.commutator(p)
raises(ValueError, lambda: p.commutator(Permutation([])))
assert len(p.atoms()) == 7
assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])
assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]
assert Permutation.from_inversion_vector(p.inversion_vector()) == p
assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
== q.array_form
raises(ValueError, lambda: Permutation.from_inversion_vector([0, 2]))
assert Permutation([i for i in range(500, -1, -1)]).inversions() == 125250
s = Permutation([0, 4, 1, 3, 2])
assert s.parity() == 0
_ = s.cyclic_form # needed to create a value for _cyclic_form
assert len(s._cyclic_form) != s.size and s.parity() == 0
assert not s.is_odd
assert s.is_even
assert Permutation([0, 1, 4, 3, 2]).parity() == 1
assert _af_parity([0, 4, 1, 3, 2]) == 0
assert _af_parity([0, 1, 4, 3, 2]) == 1
s = Permutation([0])
assert s.is_Singleton
assert Permutation([]).is_Empty
r = Permutation([3, 2, 1, 0])
assert (r**2).is_Identity
assert rmul(~p, p).is_Identity
assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
assert ~(r**2).is_Identity
assert p.max() == 6
assert p.min() == 0
q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
assert q.max() == 4
assert q.min() == 0
p = Permutation([1, 5, 2, 0, 3, 6, 4])
q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
assert p.ascents() == [0, 3, 4]
assert q.ascents() == [1, 2, 4]
assert r.ascents() == []
assert p.descents() == [1, 2, 5]
assert q.descents() == [0, 3, 5]
assert Permutation(r.descents()).is_Identity
assert p.inversions() == 7
# test the merge-sort with a longer permutation
big = list(p) + list(range(p.max() + 1, p.max() + 130))
assert Permutation(big).inversions() == 7
assert p.signature() == -1
assert q.inversions() == 11
assert q.signature() == -1
assert rmul(p, ~p).inversions() == 0
assert rmul(p, ~p).signature() == 1
assert p.order() == 6
assert q.order() == 10
assert (p**(p.order())).is_Identity
assert p.length() == 6
assert q.length() == 7
assert r.length() == 4
assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
assert r.runs() == [[3], [2], [1], [0]]
assert p.index() == 8
assert q.index() == 8
assert r.index() == 3
assert p.get_precedence_distance(q) == q.get_precedence_distance(p)
assert p.get_adjacency_distance(q) == p.get_adjacency_distance(q)
assert p.get_positional_distance(q) == p.get_positional_distance(q)
p = Permutation([0, 1, 2, 3])
q = Permutation([3, 2, 1, 0])
assert p.get_precedence_distance(q) == 6
assert p.get_adjacency_distance(q) == 3
assert p.get_positional_distance(q) == 8
p = Permutation([0, 3, 1, 2, 4])
q = Permutation.josephus(4, 5, 2)
assert p.get_adjacency_distance(q) == 3
raises(ValueError, lambda: p.get_adjacency_distance(Permutation([])))
raises(ValueError, lambda: p.get_positional_distance(Permutation([])))
raises(ValueError, lambda: p.get_precedence_distance(Permutation([])))
a = [Permutation.unrank_nonlex(4, i) for i in range(5)]
iden = Permutation([0, 1, 2, 3])
for i in range(5):
for j in range(i + 1, 5):
assert a[i].commutes_with(a[j]) == \
(rmul(a[i], a[j]) == rmul(a[j], a[i]))
if a[i].commutes_with(a[j]):
assert a[i].commutator(a[j]) == iden
assert a[j].commutator(a[i]) == iden
a = Permutation(3)
b = Permutation(0, 6, 3)(1, 2)
assert a.cycle_structure == {1: 4}
assert b.cycle_structure == {2: 1, 3: 1, 1: 2}
def test_josephus():
assert Permutation.josephus(4, 6, 1) == Permutation([3, 1, 0, 2, 5, 4])
assert Permutation.josephus(1, 5, 1).is_Identity