def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
A = []
B = rotate_right(A, 1)
assert B == []
B.append(1)
assert A == []
B = rotate_left(A, 1)
assert B == []
B.append(1)
assert A == []
def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
A = []
B = rotate_right(A, 1)
assert B == []
B.append(1)
assert A == []
B = rotate_left(A, 1)
assert B == []
B.append(1)
assert A == []
def array_form(self):
"""
This is used to convert from cyclic notation to the
canonical notation.
Currently singleton cycles need to be written
explicitly.
Examples:
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([[2,0],[3,1]])
>>> p.array_form
[1, 3, 0, 2]
"""
if self._array_form is not None:
return self._array_form
if not isinstance(self.args[0][0], list):
self._array_form = self.args[0]
return self._array_form
cycles = self.args[0]
linear_form = []
for cycle in cycles:
min_element = min(cycle)
while cycle[0] != min_element:
cycle = rotate_left(cycle, 1)
linear_form.append(cycle)
linear_form.sort(key=lambda t: -t[0])
self._array_form = list(itertools.chain(*linear_form))
return self._array_form
def dihedral(n):
"""
Generates the dihedral group of order 2n, Dn.
The result is given as a subgroup of Sn, except for the special cases n=1
(the group S2) and n=2 (the Klein 4-group) where that's not possible
and embeddings in S2 and S4 respectively are given.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.generators import dihedral
>>> list(dihedral(3))
[(2), (0 2), (0 1 2), (1 2), (0 2 1), (2)(0 1)]
See Also
========
cyclic
"""
if n == 1:
yield Permutation([0, 1])
yield Permutation([1, 0])
elif n == 2:
yield Permutation([0, 1, 2, 3])
yield Permutation([1, 0, 3, 2])
yield Permutation([2, 3, 0, 1])
yield Permutation([3, 2, 1, 0])
else:
gen = list(range(n))
for i in range(n):
yield Permutation(gen)
yield Permutation(gen[::-1])
gen = rotate_left(gen, 1)
def dihedral(n):
"""
Generates the dihedral group of order 2n, Dn.
The result is given as a subgroup of Sn, except for the special cases n=1
(the group S2) and n=2 (the Klein 4-group) where that's not possible
and embeddings in S2 and S4 respectively are given.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import dihedral
>>> list(dihedral(3))
[(2), (0 2), (0 1 2), (1 2), (0 2 1), (2)(0 1)]
See Also
========
cyclic
"""
if n == 1:
yield Permutation([0, 1])
yield Permutation([1, 0])
elif n == 2:
yield Permutation([0, 1, 2, 3])
yield Permutation([1, 0, 3, 2])
yield Permutation([2, 3, 0, 1])
yield Permutation([3, 2, 1, 0])
else:
gen = list(range(n))
for i in range(n):
yield Permutation(gen)
yield Permutation(gen[::-1])
gen = rotate_left(gen, 1)
def array_form(self):
"""
This is used to convert from cyclic notation to the
canonical notation.
Currently singleton cycles need to be written
explicitly.
Examples:
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([[2,0],[3,1]])
>>> p.array_form
[1, 3, 0, 2]
"""
if self._array_form is not None:
return self._array_form
if not isinstance(self.args[0][0], list):
self._array_form = self.args[0]
return self._array_form
cycles = self.args[0]
linear_form = []
for cycle in cycles:
min_element = min(cycle)
while cycle[0] != min_element:
cycle = rotate_left(cycle, 1)
linear_form.append(cycle)
linear_form.sort(key=lambda t: -t[0])
self._array_form = list(itertools.chain(*linear_form))
return self._array_form
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples:
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[Permutation([0, 1, 2, 3, 4]), Permutation([1, 2, 3, 4, 0]), \
Permutation([2, 3, 4, 0, 1]), Permutation([3, 4, 0, 1, 2]), \
Permutation([4, 0, 1, 2, 3])]
"""
gen = range(n)
for i in xrange(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples
========
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[Permutation([0, 1, 2, 3, 4]), Permutation([1, 2, 3, 4, 0]), \
Permutation([2, 3, 4, 0, 1]), Permutation([3, 4, 0, 1, 2]), \
Permutation([4, 0, 1, 2, 3])]
"""
gen = range(n)
for i in xrange(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def dihedral(n):
"""
Generates the dihedral group of order 2n, D2n.
Examples:
>>> from sympy.combinatorics.generators import dihedral
>>> list(dihedral(4))
[Permutation([0, 1, 2, 3]), Permutation([3, 2, 1, 0]), \
Permutation([1, 2, 3, 0]), Permutation([0, 3, 2, 1]), \
Permutation([2, 3, 0, 1]), Permutation([1, 0, 3, 2]), \
Permutation([3, 0, 1, 2]), Permutation([2, 1, 0, 3])]
"""
gen = range(n)
for i in xrange(n):
yield Permutation(gen)
yield Permutation(gen[::-1])
gen = rotate_left(gen, 1)
def dihedral(n):
"""
Generates the dihedral group of order 2n, D2n.
Examples
========
>>> from sympy.combinatorics.generators import dihedral
>>> list(dihedral(4))
[Permutation([0, 1, 2, 3]), Permutation([3, 2, 1, 0]), \
Permutation([1, 2, 3, 0]), Permutation([0, 3, 2, 1]), \
Permutation([2, 3, 0, 1]), Permutation([1, 0, 3, 2]), \
Permutation([3, 0, 1, 2]), Permutation([2, 1, 0, 3])]
"""
gen = range(n)
for i in xrange(n):
yield Permutation(gen)
yield Permutation(gen[::-1])
gen = rotate_left(gen, 1)
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples
========
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[(4), (0 1 2 3 4), (0 2 4 1 3),
(0 3 1 4 2), (0 4 3 2 1)]
See Also
========
dihedral
"""
gen = list(range(n))
for i in range(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[Permutation(4), Permutation(0, 1, 2, 3, 4), Permutation(0, 2, 4, 1, 3),
Permutation(0, 3, 1, 4, 2), Permutation(0, 4, 3, 2, 1)]
See Also
========
dihedral
"""
gen = range(n)
for i in xrange(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[Permutation(4), Permutation(0, 1, 2, 3, 4), Permutation(0, 2, 4, 1, 3),
Permutation(0, 3, 1, 4, 2), Permutation(0, 4, 3, 2, 1)]
See Also
========
dihedral
"""
gen = list(range(n))
for i in range(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def cyclic(n):
"""
Generates the cyclic group of order n, Cn.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import cyclic
>>> list(cyclic(5))
[(4), (0 1 2 3 4), (0 2 4 1 3),
(0 3 1 4 2), (0 4 3 2 1)]
See Also
========
dihedral
"""
gen = list(range(n))
for i in range(n):
yield Permutation(gen)
gen = rotate_left(gen, 1)
def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
