Subsets

class sympy.combinatorics.subsets.Subset

Represents a basic subset object.

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.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.next_binary().subset
['b']
>>> a.prev_binary().subset
['c']

classmethod bitlist_from_subset(subset, superset)

Gets the bitlist corresponding to a subset.

See also

subset_from_bitlist

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> Subset.bitlist_from_subset(['c','d'], ['a','b','c','d'])
'0011'

cardinality

Returns the number of all possible subsets.

See also

subset, superset, size, superset_size

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.cardinality
16

iterate_binary(k)

This is a helper function. It iterates over the binary subsets by k steps. This variable can be both positive or negative.

See also

next_binary, prev_binary

Examples

>>> from sympy.combinatorics.subsets 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
[]

iterate_graycode(k)

Helper function used for prev_gray and next_gray. It performs k step overs to get the respective Gray codes.

See also

next_gray, prev_gray

Examples

>>> from sympy.combinatorics.subsets 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]

next_binary()

Generates the next binary ordered subset.

See also

prev_binary, iterate_binary

Examples

>>> from sympy.combinatorics.subsets 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
[]

next_gray()

Generates the next Gray code ordered subset.

See also

iterate_graycode, prev_gray

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([1,2,3], [1,2,3,4])
>>> a.next_gray().subset
[1, 3]

next_lexicographic()

Generates the next lexicographically ordered subset.

See also

prev_lexicographic

Examples

>>> from sympy.combinatorics.subsets 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
[]

prev_binary()

Generates the previous binary ordered subset.

See also

next_binary, iterate_binary

Examples

>>> from sympy.combinatorics.subsets 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']

prev_gray()

Generates the previous Gray code ordered subset.

See also

iterate_graycode, next_gray

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([2,3,4], [1,2,3,4,5])
>>> a.prev_gray().subset
[2, 3, 4, 5]

prev_lexicographic()

Generates the previous lexicographically ordered subset.

See also

next_lexicographic

Examples

>>> from sympy.combinatorics.subsets 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']

rank_binary

Computes the binary ordered rank.

See also

iterate_binary, unrank_binary

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a','b','c','d'])
>>> a.rank_binary
0
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.rank_binary
3

rank_gray

Computes the Gray code ranking of the subset.

See also

iterate_graycode, unrank_gray

Examples

>>> from sympy.combinatorics.subsets 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

rank_lexicographic

Computes the lexicographic ranking of the subset.

Examples

>>> from sympy.combinatorics.subsets 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

size

Gets the size of the subset.

See also

subset, superset, superset_size, cardinality

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.size
2

subset

Gets the subset represented by the current instance.

See also

superset, size, superset_size, cardinality

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.subset
['c', 'd']

classmethod subset_from_bitlist(super_set, bitlist)

Gets the subset defined by the bitlist.

See also

bitlist_from_subset

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> Subset.subset_from_bitlist(['a','b','c','d'], '0011').subset
['c', 'd']

classmethod subset_indices(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)
[]

superset

Gets the superset of the subset.

See also

subset, size, superset_size, cardinality

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.superset
['a', 'b', 'c', 'd']

superset_size

Returns the size of the superset.

See also

subset, superset, size, cardinality

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.superset_size
4

classmethod unrank_binary(rank, superset)

Gets the binary ordered subset of the specified rank.

See also

iterate_binary, rank_binary

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_binary(4, ['a','b','c','d']).subset
['b']

classmethod unrank_gray(rank, superset)

Gets the Gray code ordered subset of the specified rank.

See also

iterate_graycode, rank_gray

Examples

>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_gray(4, ['a','b','c']).subset
['a', 'b']
>>> Subset.unrank_gray(0, ['a','b','c']).subset
[]

static subsets.ksubsets(superset, k)

Finds the subsets of size k in lexicographic order.

This uses the itertools generator.

See also

class

Subset

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)]