620 lines
16 KiB
Python
620 lines
16 KiB
Python
|
from itertools import combinations
|
||
|
|
|
||
|
|
from sympy.combinatorics.graycode import GrayCode
|
||
|
|
|
||
|
|
|
||
|
|
class Subset():
|
||
|
|
"""
|
||
|
|
Represents a basic subset object.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
We generate subsets using essentially two techniques,
|
||
|
|
binary enumeration and lexicographic enumeration.
|
||
|
|
The Subset class takes two arguments, the first one
|
||
|
|
describes the initial subset to consider and the second
|
||
|
|
describes the superset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.next_binary().subset
|
||
|
|
['b']
|
||
|
|
>>> a.prev_binary().subset
|
||
|
|
['c']
|
||
|
|
"""
|
||
|
|
|
||
|
|
_rank_binary = None
|
||
|
|
_rank_lex = None
|
||
|
|
_rank_graycode = None
|
||
|
|
_subset = None
|
||
|
|
_superset = None
|
||
|
|
|
||
|
|
def __new__(cls, subset, superset):
|
||
|
|
"""
|
||
|
|
Default constructor.
|
||
|
|
|
||
|
|
It takes the ``subset`` and its ``superset`` as its parameters.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.subset
|
||
|
|
['c', 'd']
|
||
|
|
>>> a.superset
|
||
|
|
['a', 'b', 'c', 'd']
|
||
|
|
>>> a.size
|
||
|
|
2
|
||
|
|
"""
|
||
|
|
if len(subset) > len(superset):
|
||
|
|
raise ValueError('Invalid arguments have been provided. The '
|
||
|
|
'superset must be larger than the subset.')
|
||
|
|
for elem in subset:
|
||
|
|
if elem not in superset:
|
||
|
|
raise ValueError('The superset provided is invalid as it does '
|
||
|
|
'not contain the element {}'.format(elem))
|
||
|
|
obj = object.__new__(cls)
|
||
|
|
obj._subset = subset
|
||
|
|
obj._superset = superset
|
||
|
|
return obj
|
||
|
|
|
||
|
|
def __eq__(self, other):
|
||
|
|
"""Return a boolean indicating whether a == b on the basis of
|
||
|
|
whether both objects are of the class Subset and if the values
|
||
|
|
of the subset and superset attributes are the same.
|
||
|
|
"""
|
||
|
|
if not isinstance(other, Subset):
|
||
|
|
return NotImplemented
|
||
|
|
return self.subset == other.subset and self.superset == other.superset
|
||
|
|
|
||
|
|
def iterate_binary(self, k):
|
||
|
|
"""
|
||
|
|
This is a helper function. It iterates over the
|
||
|
|
binary subsets by ``k`` steps. This variable can be
|
||
|
|
both positive or negative.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.iterate_binary(-2).subset
|
||
|
|
['d']
|
||
|
|
>>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.iterate_binary(2).subset
|
||
|
|
[]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
next_binary, prev_binary
|
||
|
|
"""
|
||
|
|
bin_list = Subset.bitlist_from_subset(self.subset, self.superset)
|
||
|
|
n = (int(''.join(bin_list), 2) + k) % 2**self.superset_size
|
||
|
|
bits = bin(n)[2:].rjust(self.superset_size, '0')
|
||
|
|
return Subset.subset_from_bitlist(self.superset, bits)
|
||
|
|
|
||
|
|
def next_binary(self):
|
||
|
|
"""
|
||
|
|
Generates the next binary ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.next_binary().subset
|
||
|
|
['b']
|
||
|
|
>>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.next_binary().subset
|
||
|
|
[]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
prev_binary, iterate_binary
|
||
|
|
"""
|
||
|
|
return self.iterate_binary(1)
|
||
|
|
|
||
|
|
def prev_binary(self):
|
||
|
|
"""
|
||
|
|
Generates the previous binary ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.prev_binary().subset
|
||
|
|
['a', 'b', 'c', 'd']
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.prev_binary().subset
|
||
|
|
['c']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
next_binary, iterate_binary
|
||
|
|
"""
|
||
|
|
return self.iterate_binary(-1)
|
||
|
|
|
||
|
|
def next_lexicographic(self):
|
||
|
|
"""
|
||
|
|
Generates the next lexicographically ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.next_lexicographic().subset
|
||
|
|
['d']
|
||
|
|
>>> a = Subset(['d'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.next_lexicographic().subset
|
||
|
|
[]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
prev_lexicographic
|
||
|
|
"""
|
||
|
|
i = self.superset_size - 1
|
||
|
|
indices = Subset.subset_indices(self.subset, self.superset)
|
||
|
|
|
||
|
|
if i in indices:
|
||
|
|
if i - 1 in indices:
|
||
|
|
indices.remove(i - 1)
|
||
|
|
else:
|
||
|
|
indices.remove(i)
|
||
|
|
i = i - 1
|
||
|
|
while i >= 0 and i not in indices:
|
||
|
|
i = i - 1
|
||
|
|
if i >= 0:
|
||
|
|
indices.remove(i)
|
||
|
|
indices.append(i+1)
|
||
|
|
else:
|
||
|
|
while i not in indices and i >= 0:
|
||
|
|
i = i - 1
|
||
|
|
indices.append(i + 1)
|
||
|
|
|
||
|
|
ret_set = []
|
||
|
|
super_set = self.superset
|
||
|
|
for i in indices:
|
||
|
|
ret_set.append(super_set[i])
|
||
|
|
return Subset(ret_set, super_set)
|
||
|
|
|
||
|
|
def prev_lexicographic(self):
|
||
|
|
"""
|
||
|
|
Generates the previous lexicographically ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.prev_lexicographic().subset
|
||
|
|
['d']
|
||
|
|
>>> a = Subset(['c','d'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.prev_lexicographic().subset
|
||
|
|
['c']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
next_lexicographic
|
||
|
|
"""
|
||
|
|
i = self.superset_size - 1
|
||
|
|
indices = Subset.subset_indices(self.subset, self.superset)
|
||
|
|
|
||
|
|
while i >= 0 and i not in indices:
|
||
|
|
i = i - 1
|
||
|
|
|
||
|
|
if i == 0 or i - 1 in indices:
|
||
|
|
indices.remove(i)
|
||
|
|
else:
|
||
|
|
if i >= 0:
|
||
|
|
indices.remove(i)
|
||
|
|
indices.append(i - 1)
|
||
|
|
indices.append(self.superset_size - 1)
|
||
|
|
|
||
|
|
ret_set = []
|
||
|
|
super_set = self.superset
|
||
|
|
for i in indices:
|
||
|
|
ret_set.append(super_set[i])
|
||
|
|
return Subset(ret_set, super_set)
|
||
|
|
|
||
|
|
def iterate_graycode(self, k):
|
||
|
|
"""
|
||
|
|
Helper function used for prev_gray and next_gray.
|
||
|
|
It performs ``k`` step overs to get the respective Gray codes.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
|
||
|
|
>>> a.iterate_graycode(3).subset
|
||
|
|
[1, 4]
|
||
|
|
>>> a.iterate_graycode(-2).subset
|
||
|
|
[1, 2, 4]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
next_gray, prev_gray
|
||
|
|
"""
|
||
|
|
unranked_code = GrayCode.unrank(self.superset_size,
|
||
|
|
(self.rank_gray + k) % self.cardinality)
|
||
|
|
return Subset.subset_from_bitlist(self.superset,
|
||
|
|
unranked_code)
|
||
|
|
|
||
|
|
def next_gray(self):
|
||
|
|
"""
|
||
|
|
Generates the next Gray code ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
|
||
|
|
>>> a.next_gray().subset
|
||
|
|
[1, 3]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_graycode, prev_gray
|
||
|
|
"""
|
||
|
|
return self.iterate_graycode(1)
|
||
|
|
|
||
|
|
def prev_gray(self):
|
||
|
|
"""
|
||
|
|
Generates the previous Gray code ordered subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5])
|
||
|
|
>>> a.prev_gray().subset
|
||
|
|
[2, 3, 4, 5]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_graycode, next_gray
|
||
|
|
"""
|
||
|
|
return self.iterate_graycode(-1)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def rank_binary(self):
|
||
|
|
"""
|
||
|
|
Computes the binary ordered rank.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset([], ['a','b','c','d'])
|
||
|
|
>>> a.rank_binary
|
||
|
|
0
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.rank_binary
|
||
|
|
3
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_binary, unrank_binary
|
||
|
|
"""
|
||
|
|
if self._rank_binary is None:
|
||
|
|
self._rank_binary = int("".join(
|
||
|
|
Subset.bitlist_from_subset(self.subset,
|
||
|
|
self.superset)), 2)
|
||
|
|
return self._rank_binary
|
||
|
|
|
||
|
|
@property
|
||
|
|
def rank_lexicographic(self):
|
||
|
|
"""
|
||
|
|
Computes the lexicographic ranking of the subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.rank_lexicographic
|
||
|
|
14
|
||
|
|
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
|
||
|
|
>>> a.rank_lexicographic
|
||
|
|
43
|
||
|
|
"""
|
||
|
|
if self._rank_lex is None:
|
||
|
|
def _ranklex(self, subset_index, i, n):
|
||
|
|
if subset_index == [] or i > n:
|
||
|
|
return 0
|
||
|
|
if i in subset_index:
|
||
|
|
subset_index.remove(i)
|
||
|
|
return 1 + _ranklex(self, subset_index, i + 1, n)
|
||
|
|
return 2**(n - i - 1) + _ranklex(self, subset_index, i + 1, n)
|
||
|
|
indices = Subset.subset_indices(self.subset, self.superset)
|
||
|
|
self._rank_lex = _ranklex(self, indices, 0, self.superset_size)
|
||
|
|
return self._rank_lex
|
||
|
|
|
||
|
|
@property
|
||
|
|
def rank_gray(self):
|
||
|
|
"""
|
||
|
|
Computes the Gray code ranking of the subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c','d'], ['a','b','c','d'])
|
||
|
|
>>> a.rank_gray
|
||
|
|
2
|
||
|
|
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
|
||
|
|
>>> a.rank_gray
|
||
|
|
27
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_graycode, unrank_gray
|
||
|
|
"""
|
||
|
|
if self._rank_graycode is None:
|
||
|
|
bits = Subset.bitlist_from_subset(self.subset, self.superset)
|
||
|
|
self._rank_graycode = GrayCode(len(bits), start=bits).rank
|
||
|
|
return self._rank_graycode
|
||
|
|
|
||
|
|
@property
|
||
|
|
def subset(self):
|
||
|
|
"""
|
||
|
|
Gets the subset represented by the current instance.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.subset
|
||
|
|
['c', 'd']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
superset, size, superset_size, cardinality
|
||
|
|
"""
|
||
|
|
return self._subset
|
||
|
|
|
||
|
|
@property
|
||
|
|
def size(self):
|
||
|
|
"""
|
||
|
|
Gets the size of the subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.size
|
||
|
|
2
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
subset, superset, superset_size, cardinality
|
||
|
|
"""
|
||
|
|
return len(self.subset)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def superset(self):
|
||
|
|
"""
|
||
|
|
Gets the superset of the subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.superset
|
||
|
|
['a', 'b', 'c', 'd']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
subset, size, superset_size, cardinality
|
||
|
|
"""
|
||
|
|
return self._superset
|
||
|
|
|
||
|
|
@property
|
||
|
|
def superset_size(self):
|
||
|
|
"""
|
||
|
|
Returns the size of the superset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.superset_size
|
||
|
|
4
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
subset, superset, size, cardinality
|
||
|
|
"""
|
||
|
|
return len(self.superset)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def cardinality(self):
|
||
|
|
"""
|
||
|
|
Returns the number of all possible subsets.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
>>> a.cardinality
|
||
|
|
16
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
subset, superset, size, superset_size
|
||
|
|
"""
|
||
|
|
return 2**(self.superset_size)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def subset_from_bitlist(self, super_set, bitlist):
|
||
|
|
"""
|
||
|
|
Gets the subset defined by the bitlist.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset
|
||
|
|
['c', 'd']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
bitlist_from_subset
|
||
|
|
"""
|
||
|
|
if len(super_set) != len(bitlist):
|
||
|
|
raise ValueError("The sizes of the lists are not equal")
|
||
|
|
ret_set = []
|
||
|
|
for i in range(len(bitlist)):
|
||
|
|
if bitlist[i] == '1':
|
||
|
|
ret_set.append(super_set[i])
|
||
|
|
return Subset(ret_set, super_set)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def bitlist_from_subset(self, subset, superset):
|
||
|
|
"""
|
||
|
|
Gets the bitlist corresponding to a subset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd'])
|
||
|
|
'0011'
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
subset_from_bitlist
|
||
|
|
"""
|
||
|
|
bitlist = ['0'] * len(superset)
|
||
|
|
if isinstance(subset, Subset):
|
||
|
|
subset = subset.subset
|
||
|
|
for i in Subset.subset_indices(subset, superset):
|
||
|
|
bitlist[i] = '1'
|
||
|
|
return ''.join(bitlist)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def unrank_binary(self, rank, superset):
|
||
|
|
"""
|
||
|
|
Gets the binary ordered subset of the specified rank.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset
|
||
|
|
['b']
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_binary, rank_binary
|
||
|
|
"""
|
||
|
|
bits = bin(rank)[2:].rjust(len(superset), '0')
|
||
|
|
return Subset.subset_from_bitlist(superset, bits)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def unrank_gray(self, rank, superset):
|
||
|
|
"""
|
||
|
|
Gets the Gray code ordered subset of the specified rank.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset
|
||
|
|
['a', 'b']
|
||
|
|
>>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset
|
||
|
|
[]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
iterate_graycode, rank_gray
|
||
|
|
"""
|
||
|
|
graycode_bitlist = GrayCode.unrank(len(superset), rank)
|
||
|
|
return Subset.subset_from_bitlist(superset, graycode_bitlist)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def subset_indices(self, subset, superset):
|
||
|
|
"""Return indices of subset in superset in a list; the list is empty
|
||
|
|
if all elements of ``subset`` are not in ``superset``.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import Subset
|
||
|
|
>>> superset = [1, 3, 2, 5, 4]
|
||
|
|
>>> Subset.subset_indices([3, 2, 1], superset)
|
||
|
|
[1, 2, 0]
|
||
|
|
>>> Subset.subset_indices([1, 6], superset)
|
||
|
|
[]
|
||
|
|
>>> Subset.subset_indices([], superset)
|
||
|
|
[]
|
||
|
|
|
||
|
|
"""
|
||
|
|
a, b = superset, subset
|
||
|
|
sb = set(b)
|
||
|
|
d = {}
|
||
|
|
for i, ai in enumerate(a):
|
||
|
|
if ai in sb:
|
||
|
|
d[ai] = i
|
||
|
|
sb.remove(ai)
|
||
|
|
if not sb:
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
return []
|
||
|
|
return [d[bi] for bi in b]
|
||
|
|
|
||
|
|
|
||
|
|
def ksubsets(superset, k):
|
||
|
|
"""
|
||
|
|
Finds the subsets of size ``k`` in lexicographic order.
|
||
|
|
|
||
|
|
This uses the itertools generator.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics.subsets import ksubsets
|
||
|
|
>>> list(ksubsets([1, 2, 3], 2))
|
||
|
|
[(1, 2), (1, 3), (2, 3)]
|
||
|
|
>>> list(ksubsets([1, 2, 3, 4, 5], 2))
|
||
|
|
[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), \
|
||
|
|
(2, 5), (3, 4), (3, 5), (4, 5)]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
Subset
|
||
|
|
"""
|
||
|
|
return combinations(superset, k)
|