def generate_dimino(self, af=False):
"""
yield group elements using Dimino's algorithm
If af == True it yields the array form of the permutations
Reference:
[1] The implementation of various algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
idn = range(self.degree)
order = 0
element_list = [idn]
set_element_list = set([tuple(idn)])
if af:
yield idn
else:
yield _new_from_array_form(idn)
gens = [p.array_form for p in self.generators]
for i in xrange(len(gens)):
# D elements of the subgroup G_i generated by gens[:i]
D = element_list[:]
N = [idn]
while N:
A = N
N = []
for a in A:
for g in gens[:i + 1]:
ag = perm_af_mul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = perm_af_mul(d, ag)
if af:
yield ap
else:
p = _new_from_array_form(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
def generate_dimino(self, af=False):
"""
yield group elements using Dimino's algorithm
If af == True it yields the array form of the permutations
Reference:
[1] The implementation of various algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
idn = range(self.degree)
order = 0
element_list = [idn]
set_element_list = set([tuple(idn)])
if af:
yield idn
else:
yield _new_from_array_form(idn)
gens = [p.array_form for p in self.generators]
for i in xrange(len(gens)):
# D elements of the subgroup G_i generated by gens[:i]
D = element_list[:]
N = [idn]
while N:
A = N
N = []
for a in A:
for g in gens[:i+1]:
ag = perm_af_mul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = perm_af_mul(d, ag)
if af:
yield ap
else:
p = _new_from_array_form(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
def coset_rank(self, g):
"""
rank using Schreier-Sims representation
The coset rank of `g` is the ordering number in which
it appears in the lexicographic listing according to the
coset decomposition, see coset_decomposition;
the ordering is the same as in G.generate(method='coset').
If `g` does not belong to the group it returns None
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_rank(c)
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_rank(c)
40
>>> G.coset_unrank(40, af=True)
[6, 7, 4, 5, 2, 3, 0, 1]
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
m = len(u)
a = []
un = self._coset_repr_n
n = self.degree
rank = 0
base = [1]
for i in un[m:0:-1]:
base.append(base[-1]*i)
base.reverse()
a1 = [0]*m
i1 = -1
for i in self._base:
i1 += 1
x = g1[i]
for j, h in enumerate(u[i]):
if h[i] == x:
a.append(h)
a1[i] = j
rank += j*base[i1]
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return None
if perm_af_muln(*a) == g:
return rank
return None
def coset_rank(self, g):
"""
rank using Schreier-Sims representation
The coset rank of `g` is the ordering number in which
it appears in the lexicographic listing according to the
coset decomposition, see coset_decomposition;
the ordering is the same as in G.generate(method='coset').
If `g` does not belong to the group it returns None
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_rank(c)
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_rank(c)
40
>>> G.coset_unrank(40, af=True)
[6, 7, 4, 5, 2, 3, 0, 1]
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
m = len(u)
a = []
un = self._coset_repr_n
n = self.degree
rank = 0
base = [1]
for i in un[m:0:-1]:
base.append(base[-1] * i)
base.reverse()
a1 = [0] * m
i1 = -1
for i in self._base:
i1 += 1
x = g1[i]
for j, h in enumerate(u[i]):
if h[i] == x:
a.append(h)
a1[i] = j
rank += j * base[i1]
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return None
if perm_af_muln(*a) == g:
return rank
return None
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_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 generate_schreier_sims(self, af=False):
"""
yield group elements using the Schreier-Sims representation
If af = True it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 3, 1], [0, 2, 1, 3], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
def get1(posmax):
n = len(posmax) - 1
for i in range(n,-1,-1):
if posmax[i] != 1:
return i + 1
n = self.degree
u = self.coset_repr()
# stg stack of group elements
stg = [range(n)]
# posmax[i] = len(u[i])
posmax = [len(x) for x in u]
n1 = get1(posmax)
pos = [0]*n1
posmax = posmax[:n1]
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
raise StopIteration
pos[h] = 0
h -= 1
stg.pop()
continue
p = perm_af_mul(stg[-1], u[h][pos[h]])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
yield p
else:
p1 = _new_from_array_form(p)
yield p1
stg.pop()
h -= 1
def orbit_traversal(self, alpha, af=False):
"""
compute the orbit traversal
Output: list of group elements; applying each to alpha
one gets the orbit of alpha
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.orbit_traversal(1, af=True)
[[1, 0, 2], [0, 1, 2], [0, 2, 1]]
"""
genv = [p.array_form for p in self.generators]
n = self._degree
coset_repr = [None] * n
if af:
coset_repr[alpha] = range(n)
else:
coset_repr[alpha] = Permutation(range(n))
h = 0
r = len(genv)
stg = [range(n)]
sta = [alpha]
pos = [0] * n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return coset_repr
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if coset_repr[ag] == None:
gen1 = perm_af_mul(g, stg[-1])
if af:
coset_repr[ag] = gen1
else:
coset_repr[ag] = Permutation(gen1)
sta.append(ag)
stg.append(gen1)
h += 1
def generate_schreier_sims(self, af=False):
"""
yield group elements using the Schreier-Sims representation
If af = True it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 3, 1], [0, 2, 1, 3], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
def get1(posmax):
n = len(posmax) - 1
for i in range(n, -1, -1):
if posmax[i] != 1:
return i + 1
n = self.degree
u = self.coset_repr()
# stg stack of group elements
stg = [range(n)]
# posmax[i] = len(u[i])
posmax = [len(x) for x in u]
n1 = get1(posmax)
pos = [0] * n1
posmax = posmax[:n1]
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
raise StopIteration
pos[h] = 0
h -= 1
stg.pop()
continue
p = perm_af_mul(stg[-1], u[h][pos[h]])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
yield p
else:
p1 = _new_from_array_form(p)
yield p1
stg.pop()
h -= 1
def orbit_traversal(self, alpha, af=False):
"""
compute the orbit traversal
Output: list of group elements; applying each to alpha
one gets the orbit of alpha
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.orbit_traversal(1, af=True)
[[1, 0, 2], [0, 1, 2], [0, 2, 1]]
"""
genv = [p.array_form for p in self.generators]
n = self._degree
coset_repr = [None]*n
if af:
coset_repr[alpha] = range(n)
else:
coset_repr[alpha] = Permutation(range(n))
h = 0
r = len(genv)
stg = [range(n)]
sta = [alpha]
pos = [0]*n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return coset_repr
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if coset_repr[ag] == None:
gen1 = perm_af_mul(g, stg[-1])
if af:
coset_repr[ag] = gen1
else:
coset_repr[ag] = Permutation(gen1)
sta.append(ag)
stg.append(gen1)
h += 1
def orbit(self, alpha):
"""
compute the orbit {g[i] for g in G}
It returns the orbit as a set.
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([2, 0, 1])
>>> b = Permutation([2, 1, 0])
>>> g = PermutationGroup([a, b])
>>> g.orbit(0)
set([0, 1, 2])
>>> g.orbits()
[set([0, 1, 2])]
>>> g.orbits(rep=True)
[0]
"""
genv = [p.array_form for p in self.generators]
n = self._degree
orb = set([alpha])
h = 0
r = len(genv)
stg = [range(n)]
sta = [alpha]
pos = [0]*n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return orb
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if ag not in orb:
gen1 = perm_af_mul(g, stg[-1])
orb.add(ag)
sta.append(ag)
stg.append(gen1)
h += 1
def orbit(self, alpha):
"""
compute the orbit {g[i] for g in G}
It returns the orbit as a set.
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([2, 0, 1])
>>> b = Permutation([2, 1, 0])
>>> g = PermutationGroup([a, b])
>>> g.orbit(0)
set([0, 1, 2])
>>> g.orbits()
[set([0, 1, 2])]
>>> g.orbits(rep=True)
[0]
"""
genv = [p.array_form for p in self.generators]
n = self._degree
orb = set([alpha])
h = 0
r = len(genv)
stg = [range(n)]
sta = [alpha]
pos = [0] * n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return orb
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if ag not in orb:
gen1 = perm_af_mul(g, stg[-1])
orb.add(ag)
sta.append(ag)
stg.append(gen1)
h += 1
def insert(self, g, alpha):
"""
insert permutation `g` in stabilizer chain at point alpha
"""
n = len(g)
if not g == self.idn:
vertex = self.vertex
jg = self.jg
i = _smallest_change(g, alpha)
ig = g[i]
nn = vertex[i].index_neighbor[ig]
if nn >= 0: # if ig is already neighbor of i
jginn = jg[vertex[i].perm[nn]]
if g != jginn:
# cycle consisting of two edges;
# replace jginn by g and insert h = g**-1*jginn
g1 = perm_af_invert(g)
h = perm_af_mul(g1, jginn)
jg[ vertex[i].perm[nn] ] = g
self.insert(h, alpha)
else: # new edge
self.insert_edge(g, i, ig)
self.cycle = [i]
if self.find_cycle(i, i, ig, -1):
cycle = self.cycle
cycle.append(cycle[0])
# find the smallest point (vertex) of the cycle
minn = min(cycle)
cmin = cycle.index(minn)
# now walk around the cycle starting from the smallest
# point, and multiply around the cycle to obtain h
# satisfying h[cmin] = cmin
ap = []
for c in range(cmin, len(cycle)-1) + range(cmin):
i = cycle[c]
j = cycle[c+1]
nn = vertex[i].index_neighbor[j]
p = jg[ vertex[i].perm[nn] ]
if i > j:
p = perm_af_invert(p)
ap.append(p)
ap.reverse()
h = perm_af_muln(*ap)
self.remove_edge(cycle[cmin], cycle[cmin + 1])
self.insert(h, alpha)
def coset_decomposition(self, g):
"""
Decompose `g` as h_0*...*h_{len(u)}
The Schreier-Sims coset representation u of `G`
gives a univoque decomposition of an element `g`
as h_0*...*h_{len(u)}, where h_i belongs to u[i]
Output: [h_0, .., h_{len(u)}] if `g` belongs to `G`
False otherwise
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_decomposition(c)
False
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_decomposition(c)
[[6, 4, 2, 0, 7, 5, 3, 1], [0, 4, 1, 5, 2, 6, 3, 7], [0, 1, 2, 3, 4, 5, 6, 7]]
>>> G.has_element(c)
True
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
n = len(u)
a = []
for i in range(n):
x = g1[i]
for h in u[i]:
if h[i] == x:
a.append(h)
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return False
if perm_af_muln(*a) == g:
return a
return False
def coset_decomposition(self, g):
"""
Decompose `g` as h_0*...*h_{len(u)}
The Schreier-Sims coset representation u of `G`
gives a univoque decomposition of an element `g`
as h_0*...*h_{len(u)}, where h_i belongs to u[i]
Output: [h_0, .., h_{len(u)}] if `g` belongs to `G`
False otherwise
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_decomposition(c)
False
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_decomposition(c)
[[6, 4, 2, 0, 7, 5, 3, 1], [0, 4, 1, 5, 2, 6, 3, 7], [0, 1, 2, 3, 4, 5, 6, 7]]
>>> G.has_element(c)
True
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
n = len(u)
a = []
for i in range(n):
x = g1[i]
for h in u[i]:
if h[i] == x:
a.append(h)
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return False
if perm_af_muln(*a) == g:
return a
return False
def insert(self, g, alpha):
"""
insert permutation `g` in stabilizer chain at point alpha
"""
n = len(g)
if not g == self.idn:
vertex = self.vertex
jg = self.jg
i = _smallest_change(g, alpha)
ig = g[i]
nn = vertex[i].index_neighbor[ig]
if nn >= 0: # if ig is already neighbor of i
jginn = jg[vertex[i].perm[nn]]
if g != jginn:
# cycle consisting of two edges;
# replace jginn by g and insert h = g**-1*jginn
g1 = perm_af_invert(g)
h = perm_af_mul(g1, jginn)
jg[vertex[i].perm[nn]] = g
self.insert(h, alpha)
else: # new edge
self.insert_edge(g, i, ig)
self.cycle = [i]
if self.find_cycle(i, i, ig, -1):
cycle = self.cycle
cycle.append(cycle[0])
# find the smallest point (vertex) of the cycle
minn = min(cycle)
cmin = cycle.index(minn)
# now walk around the cycle starting from the smallest
# point, and multiply around the cycle to obtain h
# satisfying h[cmin] = cmin
ap = []
for c in range(cmin, len(cycle) - 1) + range(cmin):
i = cycle[c]
j = cycle[c + 1]
nn = vertex[i].index_neighbor[j]
p = jg[vertex[i].perm[nn]]
if i > j:
p = perm_af_invert(p)
ap.append(p)
ap.reverse()
h = perm_af_muln(*ap)
self.remove_edge(cycle[cmin], cycle[cmin + 1])
self.insert(h, alpha)
def schreier_tree(self, alpha, gen):
"""
traversal of the orbit of alpha
Compute a traversal of the orbit of alpha, storing the values
in G._coset_repr; G._coset_repr[i][alpha] = i if i belongs
to the orbit of alpha.
"""
G = self.G
G._coset_repr[alpha] = gen
G._coset_repr_n += 1
genv = self.gens[:self.r]
h = 0
r = self.r
stg = [gen]
sta = [alpha]
pos = [0]*self.n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if G._coset_repr[ag] == None:
gen1 = perm_af_mul(g, stg[-1])
G._coset_repr[ag] = gen1
G._coset_repr_n += 1
sta.append(ag)
stg.append(gen1)
h += 1
def schreier_tree(self, alpha, gen):
"""
traversal of the orbit of alpha
Compute a traversal of the orbit of alpha, storing the values
in G._coset_repr; G._coset_repr[i][alpha] = i if i belongs
to the orbit of alpha.
"""
G = self.G
G._coset_repr[alpha] = gen
G._coset_repr_n += 1
genv = self.gens[:self.r]
h = 0
r = self.r
stg = [gen]
sta = [alpha]
pos = [0] * self.n
while 1:
# backtrack when finished iterating over generators
if pos[h] >= r:
if h == 0:
return
pos[h] = 0
h -= 1
sta.pop()
stg.pop()
continue
g = genv[pos[h]]
pos[h] += 1
alpha = sta[-1]
ag = g[alpha]
if G._coset_repr[ag] == None:
gen1 = perm_af_mul(g, stg[-1])
G._coset_repr[ag] = gen1
G._coset_repr_n += 1
sta.append(ag)
stg.append(gen1)
h += 1
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 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 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 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)]
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])
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
