def test_generators():
assert list(cyclic(6)) == [Permutation([0, 1, 2, 3, 4, 5]), Permutation([1, 2, 3, 4, 5, 0]), \
Permutation([2, 3, 4, 5, 0, 1]), Permutation([3, 4, 5, 0, 1, 2]), \
Permutation([4, 5, 0, 1, 2, 3]), Permutation([5, 0, 1, 2, 3, 4])]
assert list(cyclic(10)) == [Permutation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]), \
Permutation([1, 2, 3, 4, 5, 6, 7, 8, 9, 0]), \
Permutation([2, 3, 4, 5, 6, 7, 8, 9, 0, 1]), \
Permutation([3, 4, 5, 6, 7, 8, 9, 0, 1, 2]), \
Permutation([4, 5, 6, 7, 8, 9, 0, 1, 2, 3]), \
Permutation([5, 6, 7, 8, 9, 0, 1, 2, 3, 4]), \
Permutation([6, 7, 8, 9, 0, 1, 2, 3, 4, 5]), \
Permutation([7, 8, 9, 0, 1, 2, 3, 4, 5, 6]), \
Permutation([8, 9, 0, 1, 2, 3, 4, 5, 6, 7]), \
Permutation([9, 0, 1, 2, 3, 4, 5, 6, 7, 8])]
assert list(alternating(4)) == [Permutation([0, 1, 2, 3]), Permutation([0, 2, 3, 1]), \
Permutation([0, 3, 1, 2]), Permutation([1, 0, 3, 2]), \
Permutation([1, 2, 0, 3]), Permutation([1, 3, 2, 0]), \
Permutation([2, 0, 1, 3]), Permutation([2, 1, 3, 0]), \
Permutation([2, 3, 0, 1]), Permutation([3, 0, 2, 1]), \
Permutation([3, 1, 0, 2]), Permutation([3, 2, 1, 0])]
assert list(symmetric(3)) == [Permutation([0, 1, 2]), Permutation([0, 2, 1]), Permutation([1, 0, 2]), \
Permutation([1, 2, 0]), Permutation([2, 0, 1]), Permutation([2, 1, 0])]
assert list(symmetric(4)) == [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]), Permutation([0, 2, 1, 3]), \
Permutation([0, 2, 3, 1]), Permutation([0, 3, 1, 2]), Permutation([0, 3, 2, 1]), \
Permutation([1, 0, 2, 3]), Permutation([1, 0, 3, 2]), Permutation([1, 2, 0, 3]), \
Permutation([1, 2, 3, 0]), Permutation([1, 3, 0, 2]), Permutation([1, 3, 2, 0]), \
Permutation([2, 0, 1, 3]), Permutation([2, 0, 3, 1]), Permutation([2, 1, 0, 3]), \
Permutation([2, 1, 3, 0]), Permutation([2, 3, 0, 1]), Permutation([2, 3, 1, 0]), \
Permutation([3, 0, 1, 2]), Permutation([3, 0, 2, 1]), Permutation([3, 1, 0, 2]), \
Permutation([3, 1, 2, 0]), Permutation([3, 2, 0, 1]), Permutation([3, 2, 1, 0])]
assert list(dihedral(2)) == [Permutation([0, 1, 2, 3]), Permutation([1, 0, 3, 2]), \
Permutation([2, 3, 0, 1]), Permutation([3, 2, 1, 0])]
assert list(dihedral(3)) == [Permutation([0, 1, 2]), Permutation([2, 1, 0]), Permutation([1, 2, 0]), \
Permutation([0, 2, 1]), Permutation([2, 0, 1]), Permutation([1, 0, 2])]
assert list(dihedral(5)) == [Permutation([0, 1, 2, 3, 4]), Permutation([4, 3, 2, 1, 0]), \
Permutation([1, 2, 3, 4, 0]), Permutation([0, 4, 3, 2, 1]), \
Permutation([2, 3, 4, 0, 1]), Permutation([1, 0, 4, 3, 2]), \
Permutation([3, 4, 0, 1, 2]), Permutation([2, 1, 0, 4, 3]), \
Permutation([4, 0, 1, 2, 3]), Permutation([3, 2, 1, 0, 4])]
def enumerate_standard_Young_tableax(partition):
n=sum(partition)
orig = triangleOfNumbers(partition)
o=[]
for permutation in symmetric(n):
new = [[permutation(i) for i in j] for j in orig]
#new = [permutation(j) for j in orig]
if isStandard(new):
#yield new
o.append(new)
return o
def enumerate_Young_tableax_and_count_standard(partition):
n=sum(partition)
orig = triangleOfNumbers(partition)
o=[]
count=0
for permutation in symmetric(n):
new = [[permutation(i) for i in j] for j in orig]
#new = [permutation(j) for j in orig]
if isStandard(new):
count += 1
o.append(new)
return o,count
def _gl_invariant_for_identity(dim,level):
assert level % dim == 0
weight = level // dim
result = dict()
for taus in itertools.product(symmetric(dim),repeat=weight):
index = []
sign = 1
for tau in taus:
sign *= tau.signature()
for i in range(dim):
index.append( (i^tau) + 1)
result[ words.ShuffleWord(index) ] = sign
return lc.LinearCombination(result)
def _so_invariants_with_one_det(dim,level):
# no determinant fits or, using one determinant, the other spots cannot be used for inner products
if level < dim or (level - dim) % 2 == 1:
yield from []
elif level == dim:
yield _gl_invariant_for_identity(dim, dim)
else:
for det_positions in itertools.combinations(range(level),dim): # positions for the determinant
rest = set(range(level))-set(det_positions)
for partitions in equal_size_partitions(rest, 2):
inv = defaultdict(int)
for tau in symmetric(dim):
for letters in itertools.product( range(1,dim+1), repeat=level//2 ):
current_word = np.zeros( level, dtype=np.int8 )
for i in range( (level-dim)//2 ):
current_word[ partitions[i][0] ] = letters[i]
current_word[ partitions[i][1] ] = letters[i]
det_letters = [ (i^tau) + 1 for i in range(dim) ]
for i in range(dim):
current_word[ det_positions[i] ] = det_letters[i]
sign = tau.signature()
inv[ words.ShuffleWord(current_word) ] += sign
yield lc.LinearCombination( inv )
def generate_groups():
alternating_6 = list(gens.alternating(6))
odd_6 = [x for x in list(gens.symmetric(6)) if x not in alternating_6]
return alternating_6, odd_6
def getRowColumnPermutations(partition):
each_row_permutation = [list(symmetric(i)) for i in partition]
each_col_permutation = [list(symmetric(i)) for i in conjugatePartition(partition)]
return each_row_permutation,each_col_permutation
def possible_draws_without_repetition( S, n ):
"""All (ordered) sequences of drawing n times from S, without repetition."""
for subset in itertools.combinations(S,n):
for p in symmetric(n):
yield [ subset[i^p] for i in range(n) ]