def __init__(self, R, elements):
"""
Initialize ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: K = KoszulComplex(R, [x,y])
sage: TestSuite(K).run()
"""
# Generate the differentials
self._elements = elements
n = len(elements)
I = range(n)
diff = {}
zero = R.zero()
for i in I:
M = matrix(R, binomial(n,i), binomial(n,i+1), zero)
j = 0
for comb in itertools.combinations(I, i+1):
for k,val in enumerate(comb):
r = rank(comb[:k] + comb[k+1:], n, False)
M[r,j] = (-1)**k * elements[val]
j += 1
M.set_immutable()
diff[i+1] = M
diff[0] = matrix(R, 0, 1, zero)
diff[0].set_immutable()
diff[n+1] = matrix(R, 1, 0, zero)
diff[n+1].set_immutable()
ChainComplex_class.__init__(self, ZZ, ZZ(-1), R, diff)
def __init__(self, R, elements):
"""
Initialize ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: K = KoszulComplex(R, [x,y])
sage: TestSuite(K).run()
"""
# Generate the differentials
self._elements = elements
n = len(elements)
I = list(range(n))
diff = {}
zero = R.zero()
for i in I:
M = matrix(R, binomial(n, i), binomial(n, i + 1), zero)
j = 0
for comb in itertools.combinations(I, i + 1):
for k, val in enumerate(comb):
r = rank(comb[:k] + comb[k + 1:], n, False)
M[r, j] = (-1)**k * elements[val]
j += 1
M.set_immutable()
diff[i + 1] = M
diff[0] = matrix(R, 0, 1, zero)
diff[0].set_immutable()
diff[n + 1] = matrix(R, 1, 0, zero)
diff[n + 1].set_immutable()
ChainComplex_class.__init__(self, ZZ, ZZ(-1), R, diff)
def rank(comb, n, check=True):
"""
Return the rank of ``comb`` in the subsets of ``range(n)`` of size ``k``
where ``k`` is the length of ``comb``.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See: :wikipedia:`Combinadic`.
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.rank((), 3)
doctest:...: DeprecationWarning: choose_nk.rank is deprecated and will be removed. Use combination.rank instead
See http://trac.sagemath.org/18674 for details.
0
sage: choose_nk.rank((0,), 3)
0
sage: choose_nk.rank((1,), 3)
1
sage: choose_nk.rank((2,), 3)
2
sage: choose_nk.rank((0,1), 3)
0
sage: choose_nk.rank((0,2), 3)
1
sage: choose_nk.rank((1,2), 3)
2
sage: choose_nk.rank((0,1,2), 3)
0
sage: choose_nk.rank((0,1,2,3), 3)
Traceback (most recent call last):
...
ValueError: len(comb) must be <= n
sage: choose_nk.rank((0,0), 2)
Traceback (most recent call last):
...
ValueError: comb must be a subword of (0,1,...,n)
sage: choose_nk.rank([1,2], 3)
2
sage: choose_nk.rank([0,1,2], 3)
0
"""
from sage.misc.superseded import deprecation
deprecation(
18674,
"choose_nk.rank is deprecated and will be removed. Use combination.rank instead"
)
return combination.rank(comb, n, check)
def rank(comb, n, check=True):
"""
Return the rank of ``comb`` in the subsets of ``range(n)`` of size ``k``
where ``k`` is the length of ``comb``.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See :wikipedia:`Combinadic`.
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.rank((), 3)
doctest:...: DeprecationWarning: choose_nk.rank is deprecated and will be removed. Use combination.rank instead
See http://trac.sagemath.org/18674 for details.
0
sage: choose_nk.rank((0,), 3)
0
sage: choose_nk.rank((1,), 3)
1
sage: choose_nk.rank((2,), 3)
2
sage: choose_nk.rank((0,1), 3)
0
sage: choose_nk.rank((0,2), 3)
1
sage: choose_nk.rank((1,2), 3)
2
sage: choose_nk.rank((0,1,2), 3)
0
sage: choose_nk.rank((0,1,2,3), 3)
Traceback (most recent call last):
...
ValueError: len(comb) must be <= n
sage: choose_nk.rank((0,0), 2)
Traceback (most recent call last):
...
ValueError: comb must be a subword of (0,1,...,n)
sage: choose_nk.rank([1,2], 3)
2
sage: choose_nk.rank([0,1,2], 3)
0
"""
from sage.misc.superseded import deprecation
deprecation(18674, "choose_nk.rank is deprecated and will be removed. Use combination.rank instead")
return combination.rank(comb, n, check)
