from __future__ import print_function, division
from sympy.combinatorics.permutations import Permutation
from sympy.core.compatibility import range
from sympy.core.symbol import symbols
from sympy.matrices import Matrix
from sympy.utilities.iterables import variations, rotate_left
def symmetric(n):
"""
Generates the symmetric group of order n, Sn.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import symmetric
>>> list(symmetric(3))
[(2), (1 2), (2)(0 1), (0 1 2), (0 2 1), (0 2)]
"""
for perm in variations(list(range(n)), n):
yield Permutation(perm)
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 alternating(n):
"""
Generates the alternating group of order n, An.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.generators import alternating
>>> list(alternating(3))
[(2), (0 1 2), (0 2 1)]
"""
for perm in variations(list(range(n)), n):
p = Permutation(perm)
if p.is_even:
yield p
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 rubik_cube_generators():
"""Return the permutations of the 3x3 Rubik's cube, see
http://www.gap-system.org/Doc/Examples/rubik.html
"""
a = [
[(1, 3, 8, 6), (2, 5, 7, 4), (9, 33, 25, 17), (10, 34, 26, 18),
(11, 35, 27, 19)],
[(9, 11, 16, 14), (10, 13, 15, 12), (1, 17, 41, 40), (4, 20, 44, 37),
(6, 22, 46, 35)],
[(17, 19, 24, 22), (18, 21, 23, 20), (6, 25, 43, 16), (7, 28, 42, 13),
(8, 30, 41, 11)],
[(25, 27, 32, 30), (26, 29, 31, 28), (3, 38, 43, 19), (5, 36, 45, 21),
(8, 33, 48, 24)],
[(33, 35, 40, 38), (34, 37, 39, 36), (3, 9, 46, 32), (2, 12, 47, 29),
(1, 14, 48, 27)],
[(41, 43, 48, 46), (42, 45, 47, 44), (14, 22, 30, 38),
(15, 23, 31, 39), (16, 24, 32, 40)]
]
return [Permutation([[i - 1 for i in xi] for xi in x], size=48) for x in a]
def rubik(n):
"""Return permutations for an nxn Rubik's cube.
Permutations returned are for rotation of each of the slice
from the face up to the last face for each of the 3 sides (in this order):
front, right and bottom. Hence, the first n - 1 permutations are for the
slices from the front.
"""
if n < 2:
raise ValueError('dimension of cube must be > 1')
# 1-based reference to rows and columns in Matrix
def getr(f, i):
return faces[f].col(n - i)
def getl(f, i):
return faces[f].col(i - 1)
def getu(f, i):
return faces[f].row(i - 1)
def getd(f, i):
return faces[f].row(n - i)
def setr(f, i, s):
faces[f][:, n - i] = Matrix(n, 1, s)
def setl(f, i, s):
faces[f][:, i - 1] = Matrix(n, 1, s)
def setu(f, i, s):
faces[f][i - 1, :] = Matrix(1, n, s)
def setd(f, i, s):
faces[f][n - i, :] = Matrix(1, n, s)
# motion of a single face
def cw(F, r=1):
for _ in range(r):
face = faces[F]
rv = []
for c in range(n):
for r in range(n - 1, -1, -1):
rv.append(face[r, c])
faces[F] = Matrix(n, n, rv)
def ccw(F):
cw(F, 3)
# motion of plane i from the F side;
# fcw(0) moves the F face, fcw(1) moves the plane
# just behind the front face, etc...
def fcw(i, r=1):
for _ in range(r):
if i == 0:
cw(F)
i += 1
temp = getr(L, i)
setr(L, i, list((getu(D, i))))
setu(D, i, list(reversed(getl(R, i))))
setl(R, i, list((getd(U, i))))
setd(U, i, list(reversed(temp)))
i -= 1
def fccw(i):
fcw(i, 3)
# motion of the entire cube from the F side
def FCW(r=1):
for _ in range(r):
cw(F)
ccw(B)
cw(U)
t = faces[U]
cw(L)
faces[U] = faces[L]
cw(D)
faces[L] = faces[D]
cw(R)
faces[D] = faces[R]
faces[R] = t
def FCCW():
FCW(3)
# motion of the entire cube from the U side
def UCW(r=1):
for _ in range(r):
cw(U)
ccw(D)
t = faces[F]
faces[F] = faces[R]
faces[R] = faces[B]
faces[B] = faces[L]
faces[L] = t
def UCCW():
UCW(3)
# defining the permutations for the cube
U, F, R, B, L, D = names = symbols('U, F, R, B, L, D')
# the faces are represented by nxn matrices
faces = {}
count = 0
for fi in range(6):
f = []
for a in range(n**2):
f.append(count)
count += 1
faces[names[fi]] = Matrix(n, n, f)
# this will either return the value of the current permutation
# (show != 1) or else append the permutation to the group, g
def perm(show=0):
# add perm to the list of perms
p = []
for f in names:
p.extend(faces[f])
if show:
return p
g.append(Permutation(p))
g = [] # container for the group's permutations
I = list(range(6*n**2)) # the identity permutation used for checking
# define permutations corresponding to cw rotations of the planes
# up TO the last plane from that direction; by not including the
# last plane, the orientation of the cube is maintained.
# F slices
for i in range(n - 1):
fcw(i)
perm()
fccw(i) # restore
assert perm(1) == I
# R slices
# bring R to front
UCW()
for i in range(n - 1):
fcw(i)
# put it back in place
UCCW()
# record
perm()
# restore
# bring face to front
UCW()
fccw(i)
# restore
UCCW()
assert perm(1) == I
# D slices
# bring up bottom
FCW()
UCCW()
FCCW()
for i in range(n - 1):
# turn strip
fcw(i)
# put bottom back on the bottom
FCW()
UCW()
FCCW()
# record
perm()
# restore
# bring up bottom
FCW()
UCCW()
FCCW()
# turn strip
fccw(i)
# put bottom back on the bottom
FCW()
UCW()
FCCW()
assert perm(1) == I
return g
