def rank(self, sub):
"""
Returns the rank of sub as a subset of s of 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])
sage: Subsets([2,3,4],4).rank([2,3,4,5])
"""
subset = Set(sub)
lset = __builtin__.list(self.s)
lsubset = __builtin__.list(subset)
try:
index_list = sorted(map(lambda x: lset.index(x), lsubset))
except ValueError:
return None
n = len(self.s)
r = 0
if self.k not in range(len(self.s) + 1):
return None
elif self.k != len(subset):
return None
else:
return choose_nk.rank(index_list, n)
def rank(self, sub):
"""
Returns the rank of sub as 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]) is None
True
"""
subset = Set(sub)
lset = __builtin__.list(self.s)
lsubset = __builtin__.list(subset)
try:
index_list = sorted(map(lambda x: lset.index(x), lsubset))
except ValueError:
return None
n = len(self.s)
r = 0
for i in range(len(index_list)):
r += binomial(n, i)
return r + choose_nk.rank(index_list, n)
def rank(self, sub):
"""
Returns the rank of sub as a subset of s of 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])
sage: Subsets([2,3,4],4).rank([2,3,4,5])
"""
subset = Set(sub)
lset = __builtin__.list(self.s)
lsubset = __builtin__.list(subset)
try:
index_list = sorted(map(lambda x: lset.index(x), lsubset))
except ValueError:
return None
n = len(self.s)
r = 0
if self.k not in range(len(self.s)+1):
return None
elif self.k != len(subset):
return None
else:
return choose_nk.rank(index_list,n)
def rank(self, sub):
"""
Returns the rank of sub as 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]) == None
True
"""
subset = Set(sub)
lset = __builtin__.list(self.s)
lsubset = __builtin__.list(subset)
try:
index_list = sorted(map(lambda x: lset.index(x), lsubset))
except ValueError:
return None
n = len(self.s)
r = 0
for i in range(len(index_list)):
r += binomial(n,i)
return r + choose_nk.rank(index_list,n)
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 = 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)
