def yield_all_compositions_of_integer(n):
"""
Yields all compositions of positive integer ``n``.
.. container:: example
>>> for tuple_ in abjad.mathtools.yield_all_compositions_of_integer(5):
... tuple_
...
(5,)
(4, 1)
(3, 2)
(3, 1, 1)
(2, 3)
(2, 2, 1)
(2, 1, 2)
(2, 1, 1, 1)
(1, 4)
(1, 3, 1)
(1, 2, 2)
(1, 2, 1, 1)
(1, 1, 3)
(1, 1, 2, 1)
(1, 1, 1, 2)
(1, 1, 1, 1, 1)
Lists parts in descending lex order.
Parts sum to ``n``.
Finds small values of ``n`` easily.
Takes around 4 seconds for ``n`` equal to 17.
Returns integer tuple generator.
"""
from abjad import mathtools
compositions = []
integer = 0
string_length = n
while integer < 2 ** (n - 1):
binary_string = mathtools.integer_to_binary_string(integer)
binary_string = binary_string.zfill(string_length)
digits = [int(_) for _ in list(binary_string)]
partition = []
generator = itertools.groupby(digits, lambda _: _)
for value, group in generator:
partition.append(list(group))
sublengths = [len(part) for part in partition]
composition = tuple(sublengths)
compositions.append(composition)
integer += 1
for composition in reversed(sorted(compositions)):
yield composition
def yield_all_compositions_of_integer(n):
"""
Yields all compositions of positive integer ``n``.
.. container:: example
>>> for tuple_ in abjad.mathtools.yield_all_compositions_of_integer(5):
... tuple_
...
(5,)
(4, 1)
(3, 2)
(3, 1, 1)
(2, 3)
(2, 2, 1)
(2, 1, 2)
(2, 1, 1, 1)
(1, 4)
(1, 3, 1)
(1, 2, 2)
(1, 2, 1, 1)
(1, 1, 3)
(1, 1, 2, 1)
(1, 1, 1, 2)
(1, 1, 1, 1, 1)
Lists parts in descending lex order.
Parts sum to ``n``.
Finds small values of ``n`` easily.
Takes around 4 seconds for ``n`` equal to 17.
Returns integer tuple generator.
"""
from abjad import mathtools
compositions = []
integer = 0
string_length = n
while integer < 2**(n - 1):
binary_string = mathtools.integer_to_binary_string(integer)
binary_string = binary_string.zfill(string_length)
digits = [int(_) for _ in list(binary_string)]
partition = []
generator = itertools.groupby(digits, lambda _: _)
for value, group in generator:
partition.append(list(group))
sublengths = [len(part) for part in partition]
composition = tuple(sublengths)
compositions.append(composition)
integer += 1
for composition in reversed(sorted(compositions)):
yield composition
def dot_count(self):
r"""
Gets dot count.
.. container:: example
Gets dot count:
>>> for n in range(1, 16 + 1):
... try:
... duration = abjad.Duration(n, 16)
... sixteenths = duration.with_denominator(16)
... dot_count = duration.dot_count
... string = f'{sixteenths!s}\t{dot_count}'
... print(string)
... except abjad.AssignabilityError:
... sixteenths = duration.with_denominator(16)
... print(f'{sixteenths!s}\t--')
...
1/16 0
2/16 0
3/16 1
4/16 0
5/16 --
6/16 1
7/16 2
8/16 0
9/16 --
10/16 --
11/16 --
12/16 1
13/16 --
14/16 2
15/16 3
16/16 0
Dot count defined equal to number of dots required to notate duration.
Raises assignability error when duration is not assignable.
Returns positive integer.
"""
if not self.is_assignable:
raise exceptions.AssignabilityError
binary_string = mathtools.integer_to_binary_string(self.numerator)
digit_sum = sum([int(x) for x in list(binary_string)])
dot_count = digit_sum - 1
return dot_count
def yield_partitions(self):
"""
Yields partitions of sequence.
.. container:: example
>>> enumerator = abjad.Enumerator([0, 1, 2])
>>> for partition in enumerator.yield_partitions():
... partition
...
Sequence([Sequence([0, 1, 2])])
Sequence([Sequence([0, 1]), Sequence([2])])
Sequence([Sequence([0]), Sequence([1, 2])])
Sequence([Sequence([0]), Sequence([1]), Sequence([2])])
.. container:: example
>>> enumerator = abjad.Enumerator([0, 1, 2, 3])
>>> for partition in enumerator.yield_partitions():
... partition
...
Sequence([Sequence([0, 1, 2, 3])])
Sequence([Sequence([0, 1, 2]), Sequence([3])])
Sequence([Sequence([0, 1]), Sequence([2, 3])])
Sequence([Sequence([0, 1]), Sequence([2]), Sequence([3])])
Sequence([Sequence([0]), Sequence([1, 2, 3])])
Sequence([Sequence([0]), Sequence([1, 2]), Sequence([3])])
Sequence([Sequence([0]), Sequence([1]), Sequence([2, 3])])
Sequence([Sequence([0]), Sequence([1]), Sequence([2]), Sequence([3])])
Returns generator of nested sequences.
"""
length = len(self.sequence) - 1
for i in range(2 ** length):
binary_string = mathtools.integer_to_binary_string(i)
binary_string = binary_string.zfill(length)
part = list(self.sequence[0:1])
partition = [part]
for n, token in zip(self.sequence[1:], binary_string):
if int(token) == 0:
part.append(n)
else:
part = [n]
partition.append(part)
parts = [Sequence(items=_) for _ in partition]
partition = Sequence(items=parts)
yield partition
def dot_count(self) -> int:
r"""
Gets dot count.
.. container:: example
Gets dot count:
>>> for n in range(1, 16 + 1):
... try:
... duration = abjad.Duration(n, 16)
... sixteenths = duration.with_denominator(16)
... dot_count = duration.dot_count
... string = f'{sixteenths!s}\t{dot_count}'
... print(string)
... except abjad.AssignabilityError:
... sixteenths = duration.with_denominator(16)
... print(f'{sixteenths!s}\t--')
...
1/16 0
2/16 0
3/16 1
4/16 0
5/16 --
6/16 1
7/16 2
8/16 0
9/16 --
10/16 --
11/16 --
12/16 1
13/16 --
14/16 2
15/16 3
16/16 0
Dot count defined equal to number of dots required to notate duration.
Raises assignability error when duration is not assignable.
"""
if not self.is_assignable:
raise exceptions.AssignabilityError
binary_string = mathtools.integer_to_binary_string(self.numerator)
digit_sum = sum([int(x) for x in list(binary_string)])
dot_count = digit_sum - 1
return dot_count
def is_assignable_integer(argument):
r"""
Is true when ``argument`` is equivalent to an integer that can be written
without recourse to ties.
.. container:: example
>>> for n in range(0, 16 + 1):
... print('%s\t%s' % (n, abjad.mathtools.is_assignable_integer(n)))
...
0 False
1 True
2 True
3 True
4 True
5 False
6 True
7 True
8 True
9 False
10 False
11 False
12 True
13 False
14 True
15 True
16 True
Returns true or false.
"""
from abjad import mathtools
if isinstance(argument, int):
if 0 < argument:
if "01" not in mathtools.integer_to_binary_string(argument):
return True
return False
def is_assignable_integer(argument):
r"""
Is true when ``argument`` is equivalent to an integer that can be written
without recourse to ties.
.. container:: example
>>> for n in range(0, 16 + 1):
... print('%s\t%s' % (n, abjad.mathtools.is_assignable_integer(n)))
...
0 False
1 True
2 True
3 True
4 True
5 False
6 True
7 True
8 True
9 False
10 False
11 False
12 True
13 False
14 True
15 True
16 True
Returns true or false.
"""
from abjad import mathtools
if isinstance(argument, int):
if 0 < argument:
if '01' not in mathtools.integer_to_binary_string(argument):
return True
return False
def yield_combinations(self, minimum_length=None, maximum_length=None):
"""
Yields combinations of sequence items.
.. container:: example
>>> enumerator = abjad.Enumerator([1, 2, 3, 4])
>>> for combination in enumerator.yield_combinations():
... combination
Sequence([])
Sequence([1])
Sequence([2])
Sequence([1, 2])
Sequence([3])
Sequence([1, 3])
Sequence([2, 3])
Sequence([1, 2, 3])
Sequence([4])
Sequence([1, 4])
Sequence([2, 4])
Sequence([1, 2, 4])
Sequence([3, 4])
Sequence([1, 3, 4])
Sequence([2, 3, 4])
Sequence([1, 2, 3, 4])
.. container:: example
>>> enumerator = abjad.Enumerator([1, 2, 3, 4])
>>> for combination in enumerator.yield_combinations(
... minimum_length=3,
... ):
... combination
Sequence([1, 2, 3])
Sequence([1, 2, 4])
Sequence([1, 3, 4])
Sequence([2, 3, 4])
Sequence([1, 2, 3, 4])
.. container:: example
>>> enumerator = abjad.Enumerator([1, 2, 3, 4])
>>> for combination in enumerator.yield_combinations(
... maximum_length=2,
... ):
... combination
Sequence([])
Sequence([1])
Sequence([2])
Sequence([1, 2])
Sequence([3])
Sequence([1, 3])
Sequence([2, 3])
Sequence([4])
Sequence([1, 4])
Sequence([2, 4])
Sequence([3, 4])
.. container:: example
>>> enumerator = abjad.Enumerator([1, 2, 3, 4])
>>> for combination in enumerator.yield_combinations(
... minimum_length=2,
... maximum_length=2,
... ):
... combination
Sequence([1, 2])
Sequence([1, 3])
Sequence([2, 3])
Sequence([1, 4])
Sequence([2, 4])
Sequence([3, 4])
.. container:: example
>>> enumerator = abjad.Enumerator('text')
>>> for combination in enumerator.yield_combinations():
... ''.join([str(_) for _ in combination])
''
't'
'e'
'te'
'x'
'tx'
'ex'
'tex'
't'
'tt'
'et'
'tet'
'xt'
'txt'
'ext'
'text'
Yields combinations in binary string order.
Returns sequence generator.
"""
length = len(self.sequence)
for i in range(2 ** length):
binary_string = mathtools.integer_to_binary_string(i)
binary_string = binary_string.zfill(length)
sublist = []
for j, digit in enumerate(reversed(binary_string)):
if digit == "1":
sublist.append(self.sequence[j])
yield_sublist = True
if minimum_length is not None:
if len(sublist) < minimum_length:
yield_sublist = False
if maximum_length is not None:
if maximum_length < len(sublist):
yield_sublist = False
if yield_sublist:
if isinstance(self.sequence, str):
yield "".join(sublist)
else:
yield Sequence(items=sublist)
def partition_integer_into_canonic_parts(n, decrease_parts_monotonically=True):
"""
Partitions integer ``n`` into canonic parts.
.. container:: example
Returns all parts positive on positive ``n``:
>>> for n in range(1, 11):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n))
...
1 (1,)
2 (2,)
3 (3,)
4 (4,)
5 (4, 1)
6 (6,)
7 (7,)
8 (8,)
9 (8, 1)
10 (8, 2)
.. container:: example
Returns all parts negative on negative ``n``:
>>> for n in reversed(range(-20, -10)):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n))
...
-11 (-8, -3)
-12 (-12,)
-13 (-12, -1)
-14 (-14,)
-15 (-15,)
-16 (-16,)
-17 (-16, -1)
-18 (-16, -2)
-19 (-16, -3)
-20 (-16, -4)
.. container:: example
Returns parts that increase monotonically:
>>> for n in range(11, 21):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n,
... decrease_parts_monotonically=False))
...
11 (3, 8)
12 (12,)
13 (1, 12)
14 (14,)
15 (15,)
16 (16,)
17 (1, 16)
18 (2, 16)
19 (3, 16)
20 (4, 16)
Returns tuple with parts that decrease monotonically.
Returns tuple of integers.
"""
from abjad import mathtools
assert isinstance(n, int), repr(n)
assert isinstance(decrease_parts_monotonically, bool)
if n == 0:
return (0,)
result = []
previous_empty = True
binary_n = mathtools.integer_to_binary_string(abs(n))
binary_length = len(binary_n)
for i, character in enumerate(binary_n):
if character == "1":
place_value = 2 ** (binary_length - i - 1)
if previous_empty:
result.append(place_value)
else:
result[-1] += place_value
previous_empty = False
else:
previous_empty = True
sign_n = mathtools.sign(n)
if mathtools.sign(n) == -1:
result = [sign_n * _ for _ in result]
if decrease_parts_monotonically:
return tuple(result)
else:
return tuple(reversed(result))
def partition_integer_into_canonic_parts(n, decrease_parts_monotonically=True):
"""
Partitions integer ``n`` into canonic parts.
.. container:: example
Returns all parts positive on positive ``n``:
>>> for n in range(1, 11):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n))
...
1 (1,)
2 (2,)
3 (3,)
4 (4,)
5 (4, 1)
6 (6,)
7 (7,)
8 (8,)
9 (8, 1)
10 (8, 2)
.. container:: example
Returns all parts negative on negative ``n``:
>>> for n in reversed(range(-20, -10)):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n))
...
-11 (-8, -3)
-12 (-12,)
-13 (-12, -1)
-14 (-14,)
-15 (-15,)
-16 (-16,)
-17 (-16, -1)
-18 (-16, -2)
-19 (-16, -3)
-20 (-16, -4)
.. container:: example
Returns parts that increase monotonically:
>>> for n in range(11, 21):
... print(n, abjad.mathtools.partition_integer_into_canonic_parts(n,
... decrease_parts_monotonically=False))
...
11 (3, 8)
12 (12,)
13 (1, 12)
14 (14,)
15 (15,)
16 (16,)
17 (1, 16)
18 (2, 16)
19 (3, 16)
20 (4, 16)
Returns tuple with parts that decrease monotonically.
Returns tuple of integers.
"""
from abjad import mathtools
assert isinstance(n, int), repr(n)
assert isinstance(decrease_parts_monotonically, bool)
if n == 0:
return (0, )
result = []
previous_empty = True
binary_n = mathtools.integer_to_binary_string(abs(n))
binary_length = len(binary_n)
for i, character in enumerate(binary_n):
if character == '1':
place_value = 2**(binary_length - i - 1)
if previous_empty:
result.append(place_value)
else:
result[-1] += place_value
previous_empty = False
else:
previous_empty = True
sign_n = mathtools.sign(n)
if mathtools.sign(n) == -1:
result = [sign_n * _ for _ in result]
if decrease_parts_monotonically:
return tuple(result)
else:
return tuple(reversed(result))