def graded_basis(self, k):
"""
Return the basis for the ``k``-th graded piece of ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x', 3)
sage: Lyn = L.Lyndon()
sage: Lyn.graded_basis(1)
(x0, x1, x2)
sage: Lyn.graded_basis(2)
([x0, x1], [x0, x2], [x1, x2])
sage: Lyn.graded_basis(4)
([x0, [x0, [x0, x1]]],
[x0, [x0, [x0, x2]]],
[x0, [[x0, x1], x1]],
[x0, [x0, [x1, x2]]],
[x0, [[x0, x2], x1]],
[x0, [[x0, x2], x2]],
[[x0, x1], [x0, x2]],
[[[x0, x1], x1], x1],
[x0, [x1, [x1, x2]]],
[[x0, [x1, x2]], x1],
[x0, [[x1, x2], x2]],
[[[x0, x2], x1], x1],
[[x0, x2], [x1, x2]],
[[[x0, x2], x2], x1],
[[[x0, x2], x2], x2],
[x1, [x1, [x1, x2]]],
[x1, [[x1, x2], x2]],
[[[x1, x2], x2], x2])
TESTS::
sage: L = LieAlgebra(QQ, 'x,y,z', 3)
sage: Lyn = L.Lyndon()
sage: [Lyn.graded_dimension(i) for i in range(1, 11)]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
sage: [len(Lyn.graded_basis(i)) for i in range(1, 11)]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
"""
if k <= 0 or not self._indices:
return []
names = self.variable_names()
one = self.base_ring().one()
if k == 1:
return tuple(
self.element_class(self, {LieGenerator(n, k): one})
for k, n in enumerate(names))
from sage.combinat.combinat_cython import lyndon_word_iterator
n = len(self._indices)
ret = []
for lw in lyndon_word_iterator(n, k):
b = self._standard_bracket(tuple(lw))
ret.append(self.element_class(self, {b: one}))
return tuple(ret)
def graded_basis(self, k):
"""
Return the basis for the ``k``-th graded piece of ``self``.
EXAMPLES::
sage: L = LieAlgebra(QQ, 'x', 3)
sage: Lyn = L.Lyndon()
sage: Lyn.graded_basis(1)
(x0, x1, x2)
sage: Lyn.graded_basis(2)
([x0, x1], [x0, x2], [x1, x2])
sage: Lyn.graded_basis(4)
([x0, [x0, [x0, x1]]],
[x0, [x0, [x0, x2]]],
[x0, [[x0, x1], x1]],
[x0, [x0, [x1, x2]]],
[x0, [[x0, x2], x1]],
[x0, [[x0, x2], x2]],
[[x0, x1], [x0, x2]],
[[[x0, x1], x1], x1],
[x0, [x1, [x1, x2]]],
[[x0, [x1, x2]], x1],
[x0, [[x1, x2], x2]],
[[[x0, x2], x1], x1],
[[x0, x2], [x1, x2]],
[[[x0, x2], x2], x1],
[[[x0, x2], x2], x2],
[x1, [x1, [x1, x2]]],
[x1, [[x1, x2], x2]],
[[[x1, x2], x2], x2])
TESTS::
sage: L = LieAlgebra(QQ, 'x,y,z', 3)
sage: Lyn = L.Lyndon()
sage: [Lyn.graded_dimension(i) for i in range(1, 11)]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
sage: [len(Lyn.graded_basis(i)) for i in range(1, 11)]
[3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880]
"""
if k <= 0 or not self._indices:
return []
names = self.variable_names()
one = self.base_ring().one()
if k == 1:
return tuple(self.element_class(self, {LieGenerator(n, k): one})
for k,n in enumerate(names))
from sage.combinat.combinat_cython import lyndon_word_iterator
n = len(self._indices)
ret = []
for lw in lyndon_word_iterator(n, k):
b = self._standard_bracket(tuple(lw))
ret.append(self.element_class(self, {b: one}))
return tuple(ret)
def __iter__(self):
"""
TESTS::
sage: LyndonWords(3,3).list() # indirect doctest
[word: 112, word: 113, word: 122, word: 123, word: 132, word: 133, word: 223, word: 233]
sage: sum(1 for lw in LyndonWords(11, 6))
295020
sage: sum(1 for lw in LyndonWords(1000, 1))
1000
sage: sum(1 for lw in LyndonWords(1, 1000))
0
sage: list(LyndonWords(1, 1))
[word: 1]
"""
W = self._words._element_classes['list']
for lw in lyndon_word_iterator(self._n, self._k):
yield W(self._words, [i+1 for i in lw])