diff --git a/src/sage/combinat/tuple.py b/src/sage/combinat/tuple.py
index 26c1b4ec074..95ae68b040c 100644
--- a/src/sage/combinat/tuple.py
+++ b/src/sage/combinat/tuple.py
@@ -16,12 +16,12 @@
# https://www.gnu.org/licenses/
# ****************************************************************************
-from sage.libs.gap.libgap import libgap
+from sage.arith.misc import binomial
from sage.rings.integer_ring import ZZ
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
-
+from itertools import product, combinations_with_replacement
class Tuples(Parent, UniqueRepresentation):
"""
@@ -35,23 +35,23 @@ class Tuples(Parent, UniqueRepresentation):
sage: S = [1,2]
sage: Tuples(S,3).list()
- [[1, 1, 1], [2, 1, 1], [1, 2, 1], [2, 2, 1], [1, 1, 2],
- [2, 1, 2], [1, 2, 2], [2, 2, 2]]
+ [(1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1), (1, 1, 2),
+ (2, 1, 2), (1, 2, 2), (2, 2, 2)]
sage: mset = ["s","t","e","i","n"]
sage: Tuples(mset,2).list()
- [['s', 's'], ['t', 's'], ['e', 's'], ['i', 's'], ['n', 's'],
- ['s', 't'], ['t', 't'], ['e', 't'], ['i', 't'], ['n', 't'],
- ['s', 'e'], ['t', 'e'], ['e', 'e'], ['i', 'e'], ['n', 'e'],
- ['s', 'i'], ['t', 'i'], ['e', 'i'], ['i', 'i'], ['n', 'i'],
- ['s', 'n'], ['t', 'n'], ['e', 'n'], ['i', 'n'], ['n', 'n']]
+ [('s', 's'), ('t', 's'), ('e', 's'), ('i', 's'), ('n', 's'),
+ ('s', 't'), ('t', 't'), ('e', 't'), ('i', 't'), ('n', 't'),
+ ('s', 'e'), ('t', 'e'), ('e', 'e'), ('i', 'e'), ('n', 'e'),
+ ('s', 'i'), ('t', 'i'), ('e', 'i'), ('i', 'i'), ('n', 'i'),
+ ('s', 'n'), ('t', 'n'), ('e', 'n'), ('i', 'n'), ('n', 'n')]
::
sage: K.<a> = GF(4, 'a') # optional - sage.rings.finite_rings
sage: mset = [x for x in K if x != 0] # optional - sage.rings.finite_rings
sage: Tuples(mset,2).list() # optional - sage.rings.finite_rings
- [[a, a], [a + 1, a], [1, a], [a, a + 1], [a + 1, a + 1], [1, a + 1],
- [a, 1], [a + 1, 1], [1, 1]]
+ [(a, a), (a + 1, a), (1, a), (a, a + 1), (a + 1, a + 1), (1, a + 1),
+ (a, 1), (a + 1, 1), (1, 1)]
"""
@staticmethod
def __classcall_private__(cls, S, k):
@@ -75,7 +75,7 @@ def __init__(self, S, k):
"""
self.S = S
self.k = k
- self._index_list = [S.index(s) for s in S]
+ self._index_list = list(set(S.index(s) for s in S))
category = FiniteEnumeratedSets()
Parent.__init__(self, category=category)
@@ -94,33 +94,21 @@ def __iter__(self):
sage: S = [1,2]
sage: Tuples(S,3).list()
- [[1, 1, 1], [2, 1, 1], [1, 2, 1], [2, 2, 1], [1, 1, 2],
- [2, 1, 2], [1, 2, 2], [2, 2, 2]]
+ [(1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1), (1, 1, 2),
+ (2, 1, 2), (1, 2, 2), (2, 2, 2)]
sage: mset = ["s","t","e","i","n"]
sage: Tuples(mset,2).list()
- [['s', 's'], ['t', 's'], ['e', 's'], ['i', 's'], ['n', 's'],
- ['s', 't'], ['t', 't'], ['e', 't'], ['i', 't'],
- ['n', 't'], ['s', 'e'], ['t', 'e'], ['e', 'e'], ['i', 'e'],
- ['n', 'e'], ['s', 'i'], ['t', 'i'], ['e', 'i'],
- ['i', 'i'], ['n', 'i'], ['s', 'n'], ['t', 'n'], ['e', 'n'],
- ['i', 'n'], ['n', 'n']]
- """
- S = self.S
- k = self.k
- import copy
- if k <= 0:
- yield []
- return
- if k == 1:
- for x in S:
- yield [x]
- return
-
- for s in S:
- for x in Tuples(S, k - 1):
- y = copy.copy(x)
- y.append(s)
- yield y
+ [('s', 's'), ('t', 's'), ('e', 's'), ('i', 's'), ('n', 's'),
+ ('s', 't'), ('t', 't'), ('e', 't'), ('i', 't'), ('n', 't'),
+ ('s', 'e'), ('t', 'e'), ('e', 'e'), ('i', 'e'), ('n', 'e'),
+ ('s', 'i'), ('t', 'i'), ('e', 'i'), ('i', 'i'), ('n', 'i'),
+ ('s', 'n'), ('t', 'n'), ('e', 'n'), ('i', 'n'), ('n', 'n')]
+ sage: Tuples((1,1,2),3).list()
+ [(1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1), (1, 1, 2),
+ (2, 1, 2), (1, 2, 2), (2, 2, 2)]
+ """
+ for p in product(self._index_list, repeat=self.k):
+ yield tuple(self.S[i] for i in reversed(p))
def cardinality(self):
"""
@@ -133,7 +121,7 @@ def cardinality(self):
sage: Tuples(S,2).cardinality() # optional - sage.libs.gap
25
"""
- return ZZ(libgap.NrTuples(self._index_list, ZZ(self.k)))
+ return ZZ(len(self._index_list)).__pow__(self.k)
Tuples_sk = Tuples
@@ -151,10 +139,10 @@ class UnorderedTuples(Parent, UniqueRepresentation):
sage: S = [1,2]
sage: UnorderedTuples(S,3).list()
- [[1, 1, 1], [1, 1, 2], [1, 2, 2], [2, 2, 2]]
+ [(1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 2, 2)]
sage: UnorderedTuples(["a","b","c"],2).list()
- [['a', 'a'], ['a', 'b'], ['a', 'c'], ['b', 'b'], ['b', 'c'],
- ['c', 'c']]
+ [('a', 'a'), ('a', 'b'), ('a', 'c'), ('b', 'b'), ('b', 'c'),
+ ('c', 'c')]
"""
@staticmethod
def __classcall_private__(cls, S, k):
@@ -178,7 +166,7 @@ def __init__(self, S, k):
"""
self.S = S
self.k = k
- self._index_list = [S.index(s) for s in S]
+ self._index_list = list(set(S.index(s) for s in S))
category = FiniteEnumeratedSets()
Parent.__init__(self, category=category)
@@ -191,19 +179,21 @@ def __repr__(self):
"""
return "Unordered tuples of %s of length %s" % (self.S, self.k)
- def list(self):
+ def __iter__(self):
"""
EXAMPLES::
sage: S = [1,2]
sage: UnorderedTuples(S,3).list()
- [[1, 1, 1], [1, 1, 2], [1, 2, 2], [2, 2, 2]]
+ [(1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 2, 2)]
sage: UnorderedTuples(["a","b","c"],2).list()
- [['a', 'a'], ['a', 'b'], ['a', 'c'], ['b', 'b'], ['b', 'c'],
- ['c', 'c']]
+ [('a', 'a'), ('a', 'b'), ('a', 'c'), ('b', 'b'), ('b', 'c'),
+ ('c', 'c')]
+ sage: UnorderedTuples([1,1,2],3).list()
+ [(1, 1, 1), (1, 1, 2), (1, 2, 2), (2, 2, 2)]
"""
- ans = libgap.UnorderedTuples(self._index_list, ZZ(self.k))
- return [[self.S[i] for i in l] for l in ans]
+ for ans in combinations_with_replacement(self._index_list, self.k):
+ yield tuple(self.S[i] for i in ans)
def cardinality(self):
"""
@@ -213,7 +203,7 @@ def cardinality(self):
sage: UnorderedTuples(S,2).cardinality() # optional - sage.libs.gap
15
"""
- return ZZ(libgap.NrUnorderedTuples(self._index_list, ZZ(self.k)))
+ return binomial(len(self._index_list) + self.k - 1, self.k)
UnorderedTuples_sk = UnorderedTuples
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 221dca45981..744cb779fc4 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -2068,7 +2068,7 @@ def subscheme_from_Chow_form(self, Ch, dim):
L1 = []
for t in UnorderedTuples(list(range(n + 1)), dim + 1):
if all(t[i] < t[i + 1] for i in range(dim)):
- L1.append(t)
+ L1.append(list(t))
# create the dual brackets
L2 = []
signs = []
@@ -2374,7 +2374,7 @@ def rational_points(self, bound=0):
for ai in R:
P[i] = ai
for tup in S[i - 1]:
- if gcd([ai] + tup) == 1:
+ if gcd((ai,) + tup) == 1:
for j in range(i):
P[j] = tup[j]
pts.append(self(P))