Subsets¶
The set of subsets of a finite set. The set can be given as a list or a Set
or else as an integer \(n\) which encodes the set \(\{1,2,...,n\}\).
See Subsets for more information and examples.
AUTHORS:
- Mike Hansen: initial version 
- Florent Hivert (2009/02/06): doc improvements + new methods 
- class sage.combinat.subset.SubMultiset_s(s)[source]¶
- Bases: - Parent- The combinatorial class of the sub multisets of - s.- EXAMPLES: - sage: S = Subsets([1,2,2,3], submultiset=True) sage: S.cardinality() 12 sage: S.list() [[], [1], [2], [3], [1, 2], [1, 3], [2, 2], [2, 3], [1, 2, 2], [1, 2, 3], [2, 2, 3], [1, 2, 2, 3]] sage: S.first() [] sage: S.last() [1, 2, 2, 3] - >>> from sage.all import * >>> S = Subsets([Integer(1),Integer(2),Integer(2),Integer(3)], submultiset=True) >>> S.cardinality() 12 >>> S.list() [[], [1], [2], [3], [1, 2], [1, 3], [2, 2], [2, 3], [1, 2, 2], [1, 2, 3], [2, 2, 3], [1, 2, 2, 3]] >>> S.first() [] >>> S.last() [1, 2, 2, 3] - cardinality()[source]¶
- Return the cardinality of - self.- EXAMPLES: - sage: S = Subsets([1,1,2,3],submultiset=True) sage: S.cardinality() 12 sage: len(S.list()) 12 sage: S = Subsets([1,1,2,2,3],submultiset=True) sage: S.cardinality() 18 sage: len(S.list()) 18 sage: S = Subsets([1,1,1,2,2,3],submultiset=True) sage: S.cardinality() 24 sage: len(S.list()) 24 - >>> from sage.all import * >>> S = Subsets([Integer(1),Integer(1),Integer(2),Integer(3)],submultiset=True) >>> S.cardinality() 12 >>> len(S.list()) 12 >>> S = Subsets([Integer(1),Integer(1),Integer(2),Integer(2),Integer(3)],submultiset=True) >>> S.cardinality() 18 >>> len(S.list()) 18 >>> S = Subsets([Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(3)],submultiset=True) >>> S.cardinality() 24 >>> len(S.list()) 24 
 - element_class¶
- alias of - list
 - generating_serie(variable='x')[source]¶
- Return the polynomial associated to the counting of the elements of - selfweighted by the number of element they contain.- EXAMPLES: - sage: Subsets([1,1],submultiset=True).generating_serie() x^2 + x + 1 sage: Subsets([1,1,2,3],submultiset=True).generating_serie() x^4 + 3*x^3 + 4*x^2 + 3*x + 1 sage: Subsets([1,1,1,2,2,3,3,4],submultiset=True).generating_serie() x^8 + 4*x^7 + 9*x^6 + 14*x^5 + 16*x^4 + 14*x^3 + 9*x^2 + 4*x + 1 sage: S = Subsets([1,1,1,2,2,3,3,4],submultiset=True) sage: S.cardinality() 72 sage: sum(S.generating_serie()) 72 - >>> from sage.all import * >>> Subsets([Integer(1),Integer(1)],submultiset=True).generating_serie() x^2 + x + 1 >>> Subsets([Integer(1),Integer(1),Integer(2),Integer(3)],submultiset=True).generating_serie() x^4 + 3*x^3 + 4*x^2 + 3*x + 1 >>> Subsets([Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(3),Integer(3),Integer(4)],submultiset=True).generating_serie() x^8 + 4*x^7 + 9*x^6 + 14*x^5 + 16*x^4 + 14*x^3 + 9*x^2 + 4*x + 1 >>> S = Subsets([Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(3),Integer(3),Integer(4)],submultiset=True) >>> S.cardinality() 72 >>> sum(S.generating_serie()) 72 
 - random_element()[source]¶
- Return a random element of - selfwith uniform law.- EXAMPLES: - sage: S = Subsets([1,1,2,3], submultiset=True) sage: s = S.random_element() sage: s in S True - >>> from sage.all import * >>> S = Subsets([Integer(1),Integer(1),Integer(2),Integer(3)], submultiset=True) >>> s = S.random_element() >>> s in S True 
 
- class sage.combinat.subset.SubMultiset_sk(s, k)[source]¶
- Bases: - SubMultiset_s- The combinatorial class of the subsets of size - kof a multiset- s. Note that each subset is represented by a list of the elements rather than a set since we can have multiplicities (no multiset data structure yet in sage).- EXAMPLES: - sage: S = Subsets([1,2,3,3],2,submultiset=True) sage: S._k 2 sage: S.cardinality() 4 sage: S.first() [1, 2] sage: S.last() [3, 3] sage: [sub for sub in S] [[1, 2], [1, 3], [2, 3], [3, 3]] - >>> from sage.all import * >>> S = Subsets([Integer(1),Integer(2),Integer(3),Integer(3)],Integer(2),submultiset=True) >>> S._k 2 >>> S.cardinality() 4 >>> S.first() [1, 2] >>> S.last() [3, 3] >>> [sub for sub in S] [[1, 2], [1, 3], [2, 3], [3, 3]] - cardinality()[source]¶
- Return the cardinality of - self.- EXAMPLES: - sage: S = Subsets([1,2,2,3,3,3],4,submultiset=True) sage: S.cardinality() 5 sage: len(list(S)) 5 sage: S = Subsets([1,2,2,3,3,3],3,submultiset=True) sage: S.cardinality() 6 sage: len(list(S)) 6 - >>> from sage.all import * >>> S = Subsets([Integer(1),Integer(2),Integer(2),Integer(3),Integer(3),Integer(3)],Integer(4),submultiset=True) >>> S.cardinality() 5 >>> len(list(S)) 5 >>> S = Subsets([Integer(1),Integer(2),Integer(2),Integer(3),Integer(3),Integer(3)],Integer(3),submultiset=True) >>> S.cardinality() 6 >>> len(list(S)) 6 
 - generating_serie(variable='x')[source]¶
- Return the polynomial associated to the counting of the elements of - selfweighted by the number of elements they contains- EXAMPLES: - sage: x = ZZ['x'].gen() sage: l = [1,1,1,1,2,2,3] sage: for k in range(len(l)): ....: S = Subsets(l,k,submultiset=True) ....: print(S.generating_serie('x') == S.cardinality()*x**k) True True True True True True True - >>> from sage.all import * >>> x = ZZ['x'].gen() >>> l = [Integer(1),Integer(1),Integer(1),Integer(1),Integer(2),Integer(2),Integer(3)] >>> for k in range(len(l)): ... S = Subsets(l,k,submultiset=True) ... print(S.generating_serie('x') == S.cardinality()*x**k) True True True True True True True 
 - random_element()[source]¶
- Return a random submultiset of given length. - EXAMPLES: - sage: s = Subsets(7,3).random_element() sage: s in Subsets(7,3) True sage: s = Subsets(7,5).random_element() sage: s in Subsets(7,5) True - >>> from sage.all import * >>> s = Subsets(Integer(7),Integer(3)).random_element() >>> s in Subsets(Integer(7),Integer(3)) True >>> s = Subsets(Integer(7),Integer(5)).random_element() >>> s in Subsets(Integer(7),Integer(5)) True 
 
- sage.combinat.subset.Subsets(s, k=None, submultiset=False)[source]¶
- Return the combinatorial class of the subsets of the finite set - s. The set can be given as a list, Set or any iterable convertible to a set. Alternatively, a nonnegative integer \(n\) can be provided in place of- s; in this case, the result is the combinatorial class of the subsets of the set \(\{1,2,\dots,n\}\) (i.e. of the Sage- range(1,n+1)).- A second optional parameter - kcan be given. In this case,- Subsetsreturns the combinatorial class of subsets of- sof size- k.- Warning - The subsets are returned as Sets. Do not assume that these Sets are ordered; they often are not! (E.g., - Subsets(10).list()[619]returns- {10, 4, 5, 6, 7}on my system.) See- SubsetsSortedfor a similar class which returns the subsets as sorted tuples.- Finally the option - submultisetallows one to deal with sets with repeated elements, usually called multisets. The method then returns the class of all multisets in which every element is contained at most as often as it is contained in- s. These multisets are encoded as lists.- EXAMPLES: - sage: S = Subsets([1, 2, 3]); S Subsets of {1, 2, 3} sage: S.cardinality() 8 sage: S.first() {} sage: S.last() {1, 2, 3} sage: S.random_element() in S True sage: S.list() [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}] - >>> from sage.all import * >>> S = Subsets([Integer(1), Integer(2), Integer(3)]); S Subsets of {1, 2, 3} >>> S.cardinality() 8 >>> S.first() {} >>> S.last() {1, 2, 3} >>> S.random_element() in S True >>> S.list() [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}] - Here is the same example where the set is given as an integer: - sage: S = Subsets(3) sage: S.list() [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}] - >>> from sage.all import * >>> S = Subsets(Integer(3)) >>> S.list() [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}] - We demonstrate various the effect of the various options: - sage: S = Subsets(3, 2); S Subsets of {1, 2, 3} of size 2 sage: S.list() [{1, 2}, {1, 3}, {2, 3}] sage: S = Subsets([1, 2, 2], submultiset=True); S SubMultiset of [1, 2, 2] sage: S.list() [[], [1], [2], [1, 2], [2, 2], [1, 2, 2]] sage: S = Subsets([1, 2, 2, 3], 3, submultiset=True); S SubMultiset of [1, 2, 2, 3] of size 3 sage: S.list() [[1, 2, 2], [1, 2, 3], [2, 2, 3]] sage: S = Subsets(['a','b','a','b'], 2, submultiset=True); S.list() [['a', 'a'], ['a', 'b'], ['b', 'b']] - >>> from sage.all import * >>> S = Subsets(Integer(3), Integer(2)); S Subsets of {1, 2, 3} of size 2 >>> S.list() [{1, 2}, {1, 3}, {2, 3}] >>> S = Subsets([Integer(1), Integer(2), Integer(2)], submultiset=True); S SubMultiset of [1, 2, 2] >>> S.list() [[], [1], [2], [1, 2], [2, 2], [1, 2, 2]] >>> S = Subsets([Integer(1), Integer(2), Integer(2), Integer(3)], Integer(3), submultiset=True); S SubMultiset of [1, 2, 2, 3] of size 3 >>> S.list() [[1, 2, 2], [1, 2, 3], [2, 2, 3]] >>> S = Subsets(['a','b','a','b'], Integer(2), submultiset=True); S.list() [['a', 'a'], ['a', 'b'], ['b', 'b']] - And it is possible to play with subsets of subsets: - sage: S = Subsets(3) sage: S2 = Subsets(S); S2 Subsets of Subsets of {1, 2, 3} sage: S2.cardinality() 256 sage: it = iter(S2) sage: [next(it) for _ in range(8)] [{}, {{}}, {{1}}, {{2}}, {{3}}, {{1, 2}}, {{1, 3}}, {{2, 3}}] sage: S2.random_element() # random {{2}, {1, 2, 3}, {}} sage: [S2.unrank(k) for k in range(256)] == S2.list() True sage: S3 = Subsets(S2) sage: S3.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 sage: S3.unrank(14123091480) # random {{{2}, {1, 2, 3}, {1, 2}, {3}, {}}, {{1, 2, 3}, {2}, {1}, {1, 3}}, {{}, {2}, {2, 3}, {1, 2}}, {{}, {2}, {1, 2, 3}, {1, 2}}, {}, {{}, {1}, {1, 2, 3}}} sage: T = Subsets(S2, 10) sage: T.cardinality() 278826214642518400 sage: T.unrank(1441231049) # random {{{1, 2, 3}, {2}, {2, 3}}, {{3}, {1, 3}, ..., {3}, {1}, {}, {1, 3}}} - >>> from sage.all import * >>> S = Subsets(Integer(3)) >>> S2 = Subsets(S); S2 Subsets of Subsets of {1, 2, 3} >>> S2.cardinality() 256 >>> it = iter(S2) >>> [next(it) for _ in range(Integer(8))] [{}, {{}}, {{1}}, {{2}}, {{3}}, {{1, 2}}, {{1, 3}}, {{2, 3}}] >>> S2.random_element() # random {{2}, {1, 2, 3}, {}} >>> [S2.unrank(k) for k in range(Integer(256))] == S2.list() True >>> S3 = Subsets(S2) >>> S3.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 >>> S3.unrank(Integer(14123091480)) # random {{{2}, {1, 2, 3}, {1, 2}, {3}, {}}, {{1, 2, 3}, {2}, {1}, {1, 3}}, {{}, {2}, {2, 3}, {1, 2}}, {{}, {2}, {1, 2, 3}, {1, 2}}, {}, {{}, {1}, {1, 2, 3}}} >>> T = Subsets(S2, Integer(10)) >>> T.cardinality() 278826214642518400 >>> T.unrank(Integer(1441231049)) # random {{{1, 2, 3}, {2}, {2, 3}}, {{3}, {1, 3}, ..., {3}, {1}, {}, {1, 3}}} 
- class sage.combinat.subset.SubsetsSorted(s)[source]¶
- Bases: - Subsets_s- Lightweight class of all subsets of some set \(S\), with each subset being encoded as a sorted tuple. - Used to model indices of algebras given by subsets (so we don’t have to explicitly build all \(2^n\) subsets in memory). For example, - CliffordAlgebra.- element_class¶
- alias of - tuple
 - first()[source]¶
- Return the first element of - self.- EXAMPLES: - sage: from sage.combinat.subset import SubsetsSorted sage: S = SubsetsSorted(range(3)) sage: S.first() () - >>> from sage.all import * >>> from sage.combinat.subset import SubsetsSorted >>> S = SubsetsSorted(range(Integer(3))) >>> S.first() () 
 - last()[source]¶
- Return the last element of - self.- EXAMPLES: - sage: from sage.combinat.subset import SubsetsSorted sage: S = SubsetsSorted(range(3)) sage: S.last() (0, 1, 2) - >>> from sage.all import * >>> from sage.combinat.subset import SubsetsSorted >>> S = SubsetsSorted(range(Integer(3))) >>> S.last() (0, 1, 2) 
 - random_element()[source]¶
- Return a random element of - self.- EXAMPLES: - sage: from sage.combinat.subset import SubsetsSorted sage: S = SubsetsSorted(range(3)) sage: isinstance(S.random_element(), tuple) True - >>> from sage.all import * >>> from sage.combinat.subset import SubsetsSorted >>> S = SubsetsSorted(range(Integer(3))) >>> isinstance(S.random_element(), tuple) True 
 - unrank(r)[source]¶
- Return the subset which has rank - r.- EXAMPLES: - sage: from sage.combinat.subset import SubsetsSorted sage: S = SubsetsSorted(range(3)) sage: S.unrank(4) (0, 1) - >>> from sage.all import * >>> from sage.combinat.subset import SubsetsSorted >>> S = SubsetsSorted(range(Integer(3))) >>> S.unrank(Integer(4)) (0, 1) 
 
- class sage.combinat.subset.Subsets_s(s)[source]¶
- Bases: - Parent- Subsets of a given set. - EXAMPLES: - sage: S = Subsets(4); S Subsets of {1, 2, 3, 4} sage: S.cardinality() 16 sage: Subsets(4).list() [{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}] sage: S = Subsets(Subsets(Subsets(GF(3)))); S Subsets of Subsets of Subsets of Finite Field of size 3 sage: S.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 sage: S.unrank(3149254230) # random {{{1}, {0, 2}}, {{0, 1, 2}, {0, 1}, {1}, {1, 2}}, {{2}, {1, 2}, {0, 1, 2}, {0, 2}, {1}, {}}, {{1, 2}, {0}}, {{0, 1, 2}, {0, 1}, {0, 2}, {1, 2}}} - >>> from sage.all import * >>> S = Subsets(Integer(4)); S Subsets of {1, 2, 3, 4} >>> S.cardinality() 16 >>> Subsets(Integer(4)).list() [{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}] >>> S = Subsets(Subsets(Subsets(GF(Integer(3))))); S Subsets of Subsets of Subsets of Finite Field of size 3 >>> S.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 >>> S.unrank(Integer(3149254230)) # random {{{1}, {0, 2}}, {{0, 1, 2}, {0, 1}, {1}, {1, 2}}, {{2}, {1, 2}, {0, 1, 2}, {0, 2}, {1}, {}}, {{1, 2}, {0}}, {{0, 1, 2}, {0, 1}, {0, 2}, {1, 2}}} - cardinality()[source]¶
- Return the number of subsets of the set - s.- This is given by \(2^{|s|}\). - EXAMPLES: - sage: Subsets(Set([1,2,3])).cardinality() 8 sage: Subsets([1,2,3,3]).cardinality() 8 sage: Subsets(3).cardinality() 8 - >>> from sage.all import * >>> Subsets(Set([Integer(1),Integer(2),Integer(3)])).cardinality() 8 >>> Subsets([Integer(1),Integer(2),Integer(3),Integer(3)]).cardinality() 8 >>> Subsets(Integer(3)).cardinality() 8 
 - element_class[source]¶
- alias of - Set_object_enumerated
 - first()[source]¶
- Return the first subset of - s. Since we aren’t restricted to subsets of a certain size, this is always the empty set.- EXAMPLES: - sage: Subsets([1,2,3]).first() {} sage: Subsets(3).first() {} - >>> from sage.all import * >>> Subsets([Integer(1),Integer(2),Integer(3)]).first() {} >>> Subsets(Integer(3)).first() {} 
 - last()[source]¶
- Return the last subset of - s. Since we aren’t restricted to subsets of a certain size, this is always the set- sitself.- EXAMPLES: - sage: Subsets([1,2,3]).last() {1, 2, 3} sage: Subsets(3).last() {1, 2, 3} - >>> from sage.all import * >>> Subsets([Integer(1),Integer(2),Integer(3)]).last() {1, 2, 3} >>> Subsets(Integer(3)).last() {1, 2, 3} 
 - lattice()[source]¶
- Return the lattice of subsets ordered by containment. - EXAMPLES: - sage: X = Subsets([7,8,9]) sage: X.lattice() # needs sage.combinat sage.graphs Finite lattice containing 8 elements sage: Y = Subsets(0) sage: Y.lattice() # needs sage.combinat sage.graphs Finite lattice containing 1 elements - >>> from sage.all import * >>> X = Subsets([Integer(7),Integer(8),Integer(9)]) >>> X.lattice() # needs sage.combinat sage.graphs Finite lattice containing 8 elements >>> Y = Subsets(Integer(0)) >>> Y.lattice() # needs sage.combinat sage.graphs Finite lattice containing 1 elements 
 - random_element()[source]¶
- Return a random element of the class of subsets of - s(in other words, a random subset of- s).- EXAMPLES: - sage: Subsets(3).random_element() # random {2} sage: Subsets([4,5,6]).random_element() # random {5} sage: S = Subsets(Subsets(Subsets([0,1,2]))) sage: S.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 sage: s = S.random_element() sage: s # random {{{1, 2}, {2}, {0}, {1}}, {{1, 2}, {0, 1, 2}, {0, 2}, {0}, {0, 1}}, ..., {{1, 2}, {2}, {1}}, {{2}, {0, 2}, {}, {1}}} sage: s in S True - >>> from sage.all import * >>> Subsets(Integer(3)).random_element() # random {2} >>> Subsets([Integer(4),Integer(5),Integer(6)]).random_element() # random {5} >>> S = Subsets(Subsets(Subsets([Integer(0),Integer(1),Integer(2)]))) >>> S.cardinality() 115792089237316195423570985008687907853269984665640564039457584007913129639936 >>> s = S.random_element() >>> s # random {{{1, 2}, {2}, {0}, {1}}, {{1, 2}, {0, 1, 2}, {0, 2}, {0}, {0, 1}}, ..., {{1, 2}, {2}, {1}}, {{2}, {0, 2}, {}, {1}}} >>> s in S True 
 - rank(sub)[source]¶
- Return the rank of - subas a subset of- s.- EXAMPLES: - sage: Subsets(3).rank([]) 0 sage: Subsets(3).rank([1,2]) 4 sage: Subsets(3).rank([1,2,3]) 7 sage: Subsets(3).rank([2,3,4]) Traceback (most recent call last): ... ValueError: {2, 3, 4} is not a subset of {1, 2, 3} - >>> from sage.all import * >>> Subsets(Integer(3)).rank([]) 0 >>> Subsets(Integer(3)).rank([Integer(1),Integer(2)]) 4 >>> Subsets(Integer(3)).rank([Integer(1),Integer(2),Integer(3)]) 7 >>> Subsets(Integer(3)).rank([Integer(2),Integer(3),Integer(4)]) Traceback (most recent call last): ... ValueError: {2, 3, 4} is not a subset of {1, 2, 3} 
 - underlying_set()[source]¶
- Return the set of elements. - EXAMPLES: - sage: Subsets(GF(13)).underlying_set() {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} - >>> from sage.all import * >>> Subsets(GF(Integer(13))).underlying_set() {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 
 - unrank(r)[source]¶
- Return the subset of - sthat has rank- k.- EXAMPLES: - sage: Subsets(3).unrank(0) {} sage: Subsets([2,4,5]).unrank(1) {2} sage: Subsets([1,2,3]).unrank(257) Traceback (most recent call last): ... IndexError: index out of range - >>> from sage.all import * >>> Subsets(Integer(3)).unrank(Integer(0)) {} >>> Subsets([Integer(2),Integer(4),Integer(5)]).unrank(Integer(1)) {2} >>> Subsets([Integer(1),Integer(2),Integer(3)]).unrank(Integer(257)) Traceback (most recent call last): ... IndexError: index out of range 
 
- class sage.combinat.subset.Subsets_sk(s, k)[source]¶
- Bases: - Subsets_s- Subsets of fixed size of a set. - EXAMPLES: - sage: S = Subsets([0,1,2,5,7], 3); S Subsets of {0, 1, 2, 5, 7} of size 3 sage: S.cardinality() 10 sage: S.first(), S.last() ({0, 1, 2}, {2, 5, 7}) sage: S.random_element() # random {0, 5, 7} sage: S([0,2,7]) {0, 2, 7} sage: S([0,3,5]) Traceback (most recent call last): ... ValueError: {0, 3, 5} not in Subsets of {0, 1, 2, 5, 7} of size 3 sage: S([0]) Traceback (most recent call last): ... ValueError: {0} not in Subsets of {0, 1, 2, 5, 7} of size 3 - >>> from sage.all import * >>> S = Subsets([Integer(0),Integer(1),Integer(2),Integer(5),Integer(7)], Integer(3)); S Subsets of {0, 1, 2, 5, 7} of size 3 >>> S.cardinality() 10 >>> S.first(), S.last() ({0, 1, 2}, {2, 5, 7}) >>> S.random_element() # random {0, 5, 7} >>> S([Integer(0),Integer(2),Integer(7)]) {0, 2, 7} >>> S([Integer(0),Integer(3),Integer(5)]) Traceback (most recent call last): ... ValueError: {0, 3, 5} not in Subsets of {0, 1, 2, 5, 7} of size 3 >>> S([Integer(0)]) Traceback (most recent call last): ... ValueError: {0} not in Subsets of {0, 1, 2, 5, 7} of size 3 - an_element()[source]¶
- Return an example of subset. - EXAMPLES: - sage: Subsets(0,0).an_element() {} sage: Subsets(3,2).an_element() {1, 3} sage: Subsets([2,4,5],2).an_element() {2, 5} - >>> from sage.all import * >>> Subsets(Integer(0),Integer(0)).an_element() {} >>> Subsets(Integer(3),Integer(2)).an_element() {1, 3} >>> Subsets([Integer(2),Integer(4),Integer(5)],Integer(2)).an_element() {2, 5} 
 - cardinality()[source]¶
- EXAMPLES: - sage: Subsets(Set([1,2,3]), 2).cardinality() 3 sage: Subsets([1,2,3,3], 2).cardinality() 3 sage: Subsets([1,2,3], 1).cardinality() 3 sage: Subsets([1,2,3], 3).cardinality() 1 sage: Subsets([1,2,3], 0).cardinality() 1 sage: Subsets([1,2,3], 4).cardinality() 0 sage: Subsets(3,2).cardinality() 3 sage: Subsets(3,4).cardinality() 0 - >>> from sage.all import * >>> Subsets(Set([Integer(1),Integer(2),Integer(3)]), Integer(2)).cardinality() 3 >>> Subsets([Integer(1),Integer(2),Integer(3),Integer(3)], Integer(2)).cardinality() 3 >>> Subsets([Integer(1),Integer(2),Integer(3)], Integer(1)).cardinality() 3 >>> Subsets([Integer(1),Integer(2),Integer(3)], Integer(3)).cardinality() 1 >>> Subsets([Integer(1),Integer(2),Integer(3)], Integer(0)).cardinality() 1 >>> Subsets([Integer(1),Integer(2),Integer(3)], Integer(4)).cardinality() 0 >>> Subsets(Integer(3),Integer(2)).cardinality() 3 >>> Subsets(Integer(3),Integer(4)).cardinality() 0 
 - first()[source]¶
- Return the first subset of s of size k. - EXAMPLES: - sage: Subsets(Set([1,2,3]), 2).first() {1, 2} sage: Subsets([1,2,3,3], 2).first() {1, 2} sage: Subsets(3,2).first() {1, 2} sage: Subsets(3,4).first() Traceback (most recent call last): ... EmptySetError - >>> from sage.all import * >>> Subsets(Set([Integer(1),Integer(2),Integer(3)]), Integer(2)).first() {1, 2} >>> Subsets([Integer(1),Integer(2),Integer(3),Integer(3)], Integer(2)).first() {1, 2} >>> Subsets(Integer(3),Integer(2)).first() {1, 2} >>> Subsets(Integer(3),Integer(4)).first() Traceback (most recent call last): ... EmptySetError 
 - last()[source]¶
- Return the last subset of s of size k. - EXAMPLES: - sage: Subsets(Set([1,2,3]), 2).last() {2, 3} sage: Subsets([1,2,3,3], 2).last() {2, 3} sage: Subsets(3,2).last() {2, 3} sage: Subsets(3,4).last() Traceback (most recent call last): ... EmptySetError - >>> from sage.all import * >>> Subsets(Set([Integer(1),Integer(2),Integer(3)]), Integer(2)).last() {2, 3} >>> Subsets([Integer(1),Integer(2),Integer(3),Integer(3)], Integer(2)).last() {2, 3} >>> Subsets(Integer(3),Integer(2)).last() {2, 3} >>> Subsets(Integer(3),Integer(4)).last() Traceback (most recent call last): ... EmptySetError 
 - random_element()[source]¶
- Return a random element of the class of subsets of - sof size- k(in other words, a random subset of- sof size- k).- EXAMPLES: - sage: s = Subsets(3, 2).random_element() sage: s in Subsets(3, 2) True sage: Subsets(3,4).random_element() Traceback (most recent call last): ... EmptySetError - >>> from sage.all import * >>> s = Subsets(Integer(3), Integer(2)).random_element() >>> s in Subsets(Integer(3), Integer(2)) True >>> Subsets(Integer(3),Integer(4)).random_element() Traceback (most recent call last): ... EmptySetError 
 - rank(sub)[source]¶
- Return the rank of - subas a subset of- sof size- k.- EXAMPLES: - sage: Subsets(3,2).rank([1,2]) 0 sage: Subsets([2,3,4],2).rank([3,4]) 2 sage: Subsets([2,3,4],2).rank([2]) Traceback (most recent call last): ... ValueError: {2} is not a subset of length 2 of {2, 3, 4} sage: Subsets([2,3,4],4).rank([2,3,4,5]) Traceback (most recent call last): ... ValueError: {2, 3, 4, 5} is not a subset of length 4 of {2, 3, 4} - >>> from sage.all import * >>> Subsets(Integer(3),Integer(2)).rank([Integer(1),Integer(2)]) 0 >>> Subsets([Integer(2),Integer(3),Integer(4)],Integer(2)).rank([Integer(3),Integer(4)]) 2 >>> Subsets([Integer(2),Integer(3),Integer(4)],Integer(2)).rank([Integer(2)]) Traceback (most recent call last): ... ValueError: {2} is not a subset of length 2 of {2, 3, 4} >>> Subsets([Integer(2),Integer(3),Integer(4)],Integer(4)).rank([Integer(2),Integer(3),Integer(4),Integer(5)]) Traceback (most recent call last): ... ValueError: {2, 3, 4, 5} is not a subset of length 4 of {2, 3, 4} 
 - unrank(r)[source]¶
- Return the subset of - sof size- kthat has rank- r.- EXAMPLES: - sage: Subsets(3,2).unrank(0) {1, 2} sage: Subsets([2,4,5],2).unrank(0) {2, 4} sage: Subsets([1,2,8],3).unrank(42) Traceback (most recent call last): ... IndexError: index out of range - >>> from sage.all import * >>> Subsets(Integer(3),Integer(2)).unrank(Integer(0)) {1, 2} >>> Subsets([Integer(2),Integer(4),Integer(5)],Integer(2)).unrank(Integer(0)) {2, 4} >>> Subsets([Integer(1),Integer(2),Integer(8)],Integer(3)).unrank(Integer(42)) Traceback (most recent call last): ... IndexError: index out of range 
 
- sage.combinat.subset.dict_to_list(d)[source]¶
- Return a list whose elements are the elements of - iof- drepeated with multiplicity- d[i].- EXAMPLES: - sage: from sage.combinat.subset import dict_to_list sage: dict_to_list({'a':1, 'b':3}) ['a', 'b', 'b', 'b'] - >>> from sage.all import * >>> from sage.combinat.subset import dict_to_list >>> dict_to_list({'a':Integer(1), 'b':Integer(3)}) ['a', 'b', 'b', 'b'] 
- sage.combinat.subset.list_to_dict(l)[source]¶
- Return a dictionary of multiplicities and the list of its keys. - INPUT: - l– list with possibly repeated elements
 - The keys are the elements of - l(in the same order in which they appear) and values are the multiplicities of each element in- l.- EXAMPLES: - sage: from sage.combinat.subset import list_to_dict sage: list_to_dict(['a', 'b', 'b', 'b']) ({'a': 1, 'b': 3}, ['a', 'b']) - >>> from sage.all import * >>> from sage.combinat.subset import list_to_dict >>> list_to_dict(['a', 'b', 'b', 'b']) ({'a': 1, 'b': 3}, ['a', 'b']) 
- sage.combinat.subset.powerset(X)[source]¶
- Iterator over the list of all subsets of the iterable - X, in no particular order. Each list appears exactly once, up to order.- INPUT: - X– an iterable
 - OUTPUT: iterator of lists - EXAMPLES: - sage: list(powerset([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] sage: [z for z in powerset([0,[1,2]])] [[], [0], [[1, 2]], [0, [1, 2]]] - >>> from sage.all import * >>> list(powerset([Integer(1),Integer(2),Integer(3)])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] >>> [z for z in powerset([Integer(0),[Integer(1),Integer(2)]])] [[], [0], [[1, 2]], [0, [1, 2]]] - Iterating over the power set of an infinite set is also allowed: - sage: i = 0 sage: L = [] sage: for x in powerset(ZZ): ....: if i > 10: ....: break ....: else: ....: i += 1 ....: L.append(x) sage: print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2] - >>> from sage.all import * >>> i = Integer(0) >>> L = [] >>> for x in powerset(ZZ): ... if i > Integer(10): ... break ... else: ... i += Integer(1) ... L.append(x) >>> print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2] - You may also use subsets as an alias for powerset: - sage: subsets([1,2,3]) <generator object ...powerset at 0x...> sage: list(subsets([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects. - >>> from sage.all import * >>> subsets([Integer(1),Integer(2),Integer(3)]) <generator object ...powerset at 0x...> >>> list(subsets([Integer(1),Integer(2),Integer(3)])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects. - AUTHORS: - William Stein 
- Nils Bruin (2006-12-19): rewrite to work for not-necessarily finite objects X. 
 
- sage.combinat.subset.subsets(X)[source]¶
- Iterator over the list of all subsets of the iterable - X, in no particular order. Each list appears exactly once, up to order.- INPUT: - X– an iterable
 - OUTPUT: iterator of lists - EXAMPLES: - sage: list(powerset([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] sage: [z for z in powerset([0,[1,2]])] [[], [0], [[1, 2]], [0, [1, 2]]] - >>> from sage.all import * >>> list(powerset([Integer(1),Integer(2),Integer(3)])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] >>> [z for z in powerset([Integer(0),[Integer(1),Integer(2)]])] [[], [0], [[1, 2]], [0, [1, 2]]] - Iterating over the power set of an infinite set is also allowed: - sage: i = 0 sage: L = [] sage: for x in powerset(ZZ): ....: if i > 10: ....: break ....: else: ....: i += 1 ....: L.append(x) sage: print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2] - >>> from sage.all import * >>> i = Integer(0) >>> L = [] >>> for x in powerset(ZZ): ... if i > Integer(10): ... break ... else: ... i += Integer(1) ... L.append(x) >>> print(" ".join(str(x) for x in L)) [] [0] [1] [0, 1] [-1] [0, -1] [1, -1] [0, 1, -1] [2] [0, 2] [1, 2] - You may also use subsets as an alias for powerset: - sage: subsets([1,2,3]) <generator object ...powerset at 0x...> sage: list(subsets([1,2,3])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects. - >>> from sage.all import * >>> subsets([Integer(1),Integer(2),Integer(3)]) <generator object ...powerset at 0x...> >>> list(subsets([Integer(1),Integer(2),Integer(3)])) [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] The reason we return lists instead of sets is that the elements of sets must be hashable and many structures on which one wants the powerset consist of non-hashable objects. - AUTHORS: - William Stein 
- Nils Bruin (2006-12-19): rewrite to work for not-necessarily finite objects X. 
 
- sage.combinat.subset.uniq(L)[source]¶
- Iterate over the elements of - L, yielding every element at most once: keep only the first occurrence of any item.- The items must be hashable. - INPUT: - L– iterable
 - EXAMPLES: - sage: L = [1, 1, 8, -5, 3, -5, 'a', 'x', 'a'] sage: it = uniq(L); it <generator object uniq at ...> sage: list(it) [1, 8, -5, 3, 'a', 'x'] - >>> from sage.all import * >>> L = [Integer(1), Integer(1), Integer(8), -Integer(5), Integer(3), -Integer(5), 'a', 'x', 'a'] >>> it = uniq(L); it <generator object uniq at ...> >>> list(it) [1, 8, -5, 3, 'a', 'x']