def cardinality(self):
r"""
Return the number of Baxter permutations of size ``self._n``.
For any positive integer `n`, the number of Baxter
permutations of size `n` equals
.. MATH::
\sum_{k=1}^n \dfrac
{\binom{n+1}{k-1} \binom{n+1}{k} \binom{n+1}{k+1}}
{\binom{n+1}{1} \binom{n+1}{2}} .
This is :oeis:`A001181`.
EXAMPLES::
sage: [BaxterPermutations(n).cardinality() for n in range(13)]
[1, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560]
sage: BaxterPermutations(3r).cardinality()
6
sage: parent(_)
Integer Ring
"""
if self._n == 0:
return 1
from sage.arith.all import binomial
return sum((binomial(self._n + 1, k) * binomial(self._n + 1, k + 1) *
binomial(self._n + 1, k + 2)) //
((self._n + 1) * binomial(self._n + 1, 2))
for k in range(self._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 = 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 __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 cardinality(self):
r"""
Return the number of Baxter permutations of size ``self._n``.
For any positive integer `n`, the number of Baxter
permutations of size `n` equals
.. MATH::
\sum_{k=1}^n \dfrac
{\binom{n+1}{k-1} \binom{n+1}{k} \binom{n+1}{k+1}}
{\binom{n+1}{1} \binom{n+1}{2}} .
This is :oeis:`A001181`.
EXAMPLES::
sage: [BaxterPermutations(n).cardinality() for n in range(13)]
[1, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560]
sage: BaxterPermutations(3r).cardinality()
6
sage: parent(_)
Integer Ring
"""
if self._n == 0:
return 1
from sage.arith.all import binomial
return sum((binomial(self._n + 1, k) *
binomial(self._n + 1, k + 1) *
binomial(self._n + 1, k + 2)) //
((self._n + 1) * binomial(self._n + 1, 2))
for k in range(self._n))
def cardinality(self):
"""
EXAMPLES::
sage: IntegerVectors(3,3, min_part=1).cardinality()
1
sage: IntegerVectors(5,3, min_part=1).cardinality()
6
sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality()
16
"""
if not self._constraints:
if self.k < 0:
return +infinity
if self.n >= 0:
return binomial(self.n+self.k-1,self.n)
else:
return 0
else:
if len(self._constraints) == 1 and 'max_part' in self._constraints and self._constraints['max_part'] != infinity:
m = self._constraints['max_part']
if m >= self.n:
return binomial(self.n+self.k-1,self.n)
else: #do by inclusion / exclusion on the number
#i of parts greater than m
return sum( [(-1)**i * binomial(self.n+self.k-1-i*(m+1), self.k-1)*binomial(self.k,i) for i in range(0, self.n/(m+1)+1)])
else:
return super(IntegerVectors_nkconstraints, self).cardinality()
def Z2(a, b, c, check_convergence=True):
M = Multizetas(QQ)
if a == 0 and b == 0:
return M((c - 1, )) - M((c, ))
else:
return sum(
binomial(a + i - 1, i) * M((b - i, c + a + i))
for i in range(b)) + sum(
binomial(b + i - 1, i) * M((a - i, c + b + i))
for i in range(a))
def _induced_flags(self, n, tg, type_edges):
flag_counts = {}
flags = []
total = 0
for p in Tuples([0, 1], binomial(n, 2) - binomial(tg.n, 2)):
edges = list(type_edges)
c = 0
for i in range(tg.n + 1, n + 1):
for j in range(1, i):
if p[c] == 0:
edges.append((i, j))
else:
edges.append((j, i))
c += 1
ig = ThreeGraphFlag()
ig.n = n
ig.t = tg.n
for s in Combinations(list(range(1, n + 1)), 3):
if self._variant:
if ((s[1], s[0]) in edges and
(s[0], s[2]) in edges) or ((s[2], s[0]) in edges and
(s[0], s[1]) in edges):
ig.add_edge(s)
else:
if ((s[0], s[1]) in edges and (s[1], s[2]) in edges and
(s[2], s[0]) in edges) or ((s[0], s[2]) in edges and
(s[2], s[1]) in edges and
(s[1], s[0]) in edges):
ig.add_edge(s)
it = ig.induced_subgraph(list(range(1, tg.n + 1)))
if tg.is_labelled_isomorphic(it):
ig.make_minimal_isomorph()
ghash = hash(ig)
if ghash in flag_counts:
flag_counts[ghash] += 1
else:
flags.append(ig)
flag_counts[ghash] = 1
total += 1
return [(f, flag_counts[hash(f)] / Integer(total)) for f in flags]
def from_rank(r, n, k):
r"""
Returns the combination of rank ``r`` in the subsets of
``range(n)`` of size ``k`` when listed in lexicographic order.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See: :wikipedia:`Combinadic`
EXAMPLES::
sage: import sage.combinat.combination as combination
sage: combination.from_rank(0,3,0)
()
sage: combination.from_rank(0,3,1)
(0,)
sage: combination.from_rank(1,3,1)
(1,)
sage: combination.from_rank(2,3,1)
(2,)
sage: combination.from_rank(0,3,2)
(0, 1)
sage: combination.from_rank(1,3,2)
(0, 2)
sage: combination.from_rank(2,3,2)
(1, 2)
sage: combination.from_rank(0,3,3)
(0, 1, 2)
"""
if k < 0:
raise ValueError("k must be > 0")
if k > n:
raise ValueError("k must be <= n")
a = n
b = k
x = binomial(n, k) - 1 - r # x is the 'dual' of m
comb = [None] * k
for i in range(k):
comb[i] = _comb_largest(a, b, x)
x = x - binomial(comb[i], b)
a = comb[i]
b = b - 1
for i in range(k):
comb[i] = (n - 1) - comb[i]
return tuple(comb)
def from_rank(r, n, k):
r"""
Returns the combination of rank ``r`` in the subsets of
``range(n)`` of size ``k`` when listed in lexicographic order.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See: :wikipedia:`Combinadic`
EXAMPLES::
sage: import sage.combinat.combination as combination
sage: combination.from_rank(0,3,0)
()
sage: combination.from_rank(0,3,1)
(0,)
sage: combination.from_rank(1,3,1)
(1,)
sage: combination.from_rank(2,3,1)
(2,)
sage: combination.from_rank(0,3,2)
(0, 1)
sage: combination.from_rank(1,3,2)
(0, 2)
sage: combination.from_rank(2,3,2)
(1, 2)
sage: combination.from_rank(0,3,3)
(0, 1, 2)
"""
if k < 0:
raise ValueError("k must be > 0")
if k > n:
raise ValueError("k must be <= n")
a = n
b = k
x = binomial(n, k) - 1 - r # x is the 'dual' of m
comb = [None] * k
for i in xrange(k):
comb[i] = _comb_largest(a, b, x)
x = x - binomial(comb[i], b)
a = comb[i]
b = b - 1
for i in xrange(k):
comb[i] = (n - 1) - comb[i]
return tuple(comb)
def unrank(self, r):
"""
Return the subset of ``s`` that 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
"""
r = Integer(r)
if r >= self.cardinality() or r < 0:
raise IndexError("index out of range")
else:
k = ZZ_0
n = self._s.cardinality()
bin = Integer(1)
while r >= bin:
r -= bin
k += 1
bin = binomial(n,k)
return self.element_class([self._s.unrank(i) for i in combination.from_rank(r, n, k)])
def rank(self, sub):
"""
Return 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])
Traceback (most recent call last):
...
ValueError: {2, 3, 4} is not a subset of {1, 2, 3}
"""
if sub not in Sets():
ssub = Set(sub)
if len(sub) != len(ssub):
raise ValueError("repeated elements in {}".format(sub))
sub = ssub
try:
index_list = sorted(self._s.rank(x) for x in sub)
except (ValueError,IndexError):
raise ValueError("{} is not a subset of {}".format(
Set(sub), self._s))
n = self._s.cardinality()
r = sum(binomial(n,i) for i in range(len(index_list)))
return r + combination.rank(index_list,n)
def unrank(self, r):
"""
Return the subset of ``s`` that 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
"""
r = Integer(r)
if r >= self.cardinality() or r < 0:
raise IndexError("index out of range")
else:
k = ZZ_0
n = self._s.cardinality()
bin = Integer(1)
while r >= bin:
r -= bin
k += 1
bin = binomial(n, k)
return self.element_class(
[self._s.unrank(i) for i in combination.from_rank(r, n, k)])
def cardinality(self):
r"""
Return the number of words in the shuffle product
of ``w1`` and ``w2``.
This is understood as a multiset cardinality, not as a
set cardinality; it does not count the distinct words only.
It is given by `\binom{l_1+l_2}{l_1}`, where `l_1` is the
length of ``w1`` and where `l_2` is the length of ``w2``.
EXAMPLES::
sage: from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2
sage: w, u = map(Words("abcd"), ["ab", "cd"])
sage: S = ShuffleProduct_w1w2(w,u)
sage: S.cardinality()
6
sage: w, u = map(Words("ab"), ["ab", "ab"])
sage: S = ShuffleProduct_w1w2(w,u)
sage: S.cardinality()
6
"""
return binomial(self._w1.length()+self._w2.length(), self._w1.length())
def __init__(self, fmodule, degree, name=None, latex_name=None):
r"""
TESTS::
sage: from sage.tensor.modules.ext_pow_free_module import ExtPowerFreeModule
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: A = ExtPowerFreeModule(M, 2) ; A
2nd exterior power of the Rank-3 free module M over the
Integer Ring
sage: TestSuite(A).run()
"""
from sage.arith.all import binomial
from sage.typeset.unicode_characters import unicode_bigwedge
self._fmodule = fmodule
self._degree = ZZ(degree)
rank = binomial(fmodule._rank, degree)
if name is None and fmodule._name is not None:
name = unicode_bigwedge + r'^{}('.format(degree) \
+ fmodule._name + ')'
if latex_name is None and fmodule._latex_name is not None:
latex_name = r'\Lambda^{' + str(degree) + r'}\left(' \
+ fmodule._latex_name + r'\right)'
FiniteRankFreeModule.__init__(
self,
fmodule._ring,
rank,
name=name,
latex_name=latex_name,
start_index=fmodule._sindex,
output_formatter=fmodule._output_formatter)
fmodule._all_modules.add(self)
def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`.
See formula in section 9.2 of [PS2011]_
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a)))**(j - r)) *
(p**r) * symb_twisted.moment(r) for r in range(j + 1)) / ap
def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return (
sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a))) ** (j - r)) * (p ** r) * symb_twisted.moment(r)
for r in range(j + 1)
)
/ ap
)
def rank(self, x):
"""
Returns the position of a given element.
INPUT:
- ``x`` - a list with ``sum(x) == n`` and ``len(x) == k``
TESTS::
sage: IV = IntegerVectors(4,5)
sage: range(IV.cardinality()) == [IV.rank(x) for x in IV]
True
"""
if x not in self:
raise ValueError(
"argument is not a member of IntegerVectors(%d,%d)" %
(self.n, self.k))
n = self.n
k = self.k
r = 0
for i in range(k - 1):
k -= 1
n -= x[i]
r += binomial(k + n - 1, k)
return r
def log_gamma_binomial(p, gamma, z, n, M):
r"""
Return the list of coefficients in the power series
expansion (up to precision `M`) of `\binom{\log_p(z)/\log_p(\gamma)}{n}`
INPUT:
- ``p`` -- prime
- ``gamma`` -- topological generator, e.g. `1+p`
- ``z`` -- variable
- ``n`` -- nonnegative integer
- ``M`` -- precision
OUTPUT:
The list of coefficients in the power series expansion of
`\binom{\log_p(z)/\log_p(\gamma)}{n}`
EXAMPLES::
sage: R.<z> = QQ['z']
sage: from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
sage: log_gamma_binomial(5,1+5,z,2,4)
[0, -3/205, 651/84050, -223/42025]
sage: log_gamma_binomial(5,1+5,z,3,4)
[0, 2/205, -223/42025, 95228/25845375]
"""
L = sum([ZZ(-1) ** j / j * z ** j for j in range(1, M)]) # log_p(1+z)
loggam = L / (L(gamma - 1)) # log_{gamma}(1+z)= log_p(1+z)/log_p(gamma)
return z.parent()(binomial(loggam, n)).truncate(M).list()
def rank(self, x):
"""
Returns the position of a given element.
INPUT:
- ``x`` - a list with ``sum(x) == n`` and ``len(x) == k``
TESTS::
sage: IV = IntegerVectors(4,5)
sage: range(IV.cardinality()) == [IV.rank(x) for x in IV]
True
"""
if x not in self:
raise ValueError("argument is not a member of IntegerVectors(%d,%d)" % (self.n, self.k))
n = self.n
k = self.k
r = 0
for i in range(k-1):
k -= 1
n -= x[i]
r += binomial(k+n-1,k)
return r
def upper_bound(min_length, max_length, floor, ceiling, min_slope, max_slope):
"""
Compute a coarse upper bound on the size of a vector satisfying the
constraints.
TESTS::
sage: import sage.combinat.integer_list_old as integer_list
sage: f = lambda x: lambda i: x
sage: integer_list.upper_bound(0,4,f(0), f(1),-infinity,infinity)
4
sage: integer_list.upper_bound(0, infinity, f(0), f(1), -infinity, infinity)
inf
sage: integer_list.upper_bound(0, infinity, f(0), f(1), -infinity, -1)
1
sage: integer_list.upper_bound(0, infinity, f(0), f(5), -infinity, -1)
15
sage: integer_list.upper_bound(0, infinity, f(0), f(5), -infinity, -2)
9
"""
from sage.functions.all import floor as flr
if max_length < float('inf'):
return sum([ceiling(j) for j in range(max_length)])
elif max_slope < 0 and ceiling(1) < float('inf'):
maxl = flr(-ceiling(1) / max_slope)
return ceiling(1) * (maxl + 1) + binomial(maxl + 1, 2) * max_slope
#FIXME: only checking the first 10000 values, but that should generally
#be enough
elif [ceiling(j) for j in range(10000)] == [0] * 10000:
return 0
else:
return float('inf')
def upper_bound(min_length, max_length, floor, ceiling, min_slope, max_slope):
"""
Compute a coarse upper bound on the size of a vector satisfying the
constraints.
TESTS::
sage: import sage.combinat.integer_list_old as integer_list
sage: f = lambda x: lambda i: x
sage: integer_list.upper_bound(0,4,f(0), f(1),-infinity,infinity)
4
sage: integer_list.upper_bound(0, infinity, f(0), f(1), -infinity, infinity)
inf
sage: integer_list.upper_bound(0, infinity, f(0), f(1), -infinity, -1)
1
sage: integer_list.upper_bound(0, infinity, f(0), f(5), -infinity, -1)
15
sage: integer_list.upper_bound(0, infinity, f(0), f(5), -infinity, -2)
9
"""
from sage.functions.all import floor as flr
if max_length < float('inf'):
return sum( [ ceiling(j) for j in range(max_length)] )
elif max_slope < 0 and ceiling(1) < float('inf'):
maxl = flr(-ceiling(1)/max_slope)
return ceiling(1)*(maxl+1) + binomial(maxl+1,2)*max_slope
#FIXME: only checking the first 10000 values, but that should generally
#be enough
elif [ceiling(j) for j in range(10000)] == [0]*10000:
return 0
else:
return float('inf')
def log_gamma_binomial(p, gamma, z, n, M):
r"""
Return the list of coefficients in the power series
expansion (up to precision `M`) of `\binom{\log_p(z)/\log_p(\gamma)}{n}`
INPUT:
- ``p`` -- prime
- ``gamma`` -- topological generator, e.g. `1+p`
- ``z`` -- variable
- ``n`` -- nonnegative integer
- ``M`` -- precision
OUTPUT:
The list of coefficients in the power series expansion of
`\binom{\log_p(z)/\log_p(\gamma)}{n}`
EXAMPLES::
sage: R.<z> = QQ['z']
sage: from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
sage: log_gamma_binomial(5,1+5,z,2,4)
[0, -3/205, 651/84050, -223/42025]
sage: log_gamma_binomial(5,1+5,z,3,4)
[0, 2/205, -223/42025, 95228/25845375]
"""
L = sum([ZZ(-1) ** j / j * z ** j for j in range(1, M)]) # log_p(1+z)
loggam = L / (L(gamma - 1)) # log_{gamma}(1+z)= log_p(1+z)/log_p(gamma)
return z.parent()(binomial(loggam, n)).truncate(M).list()
def rank(self, sub):
"""
Return 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])
Traceback (most recent call last):
...
ValueError: {2, 3, 4} is not a subset of {1, 2, 3}
"""
if sub not in Sets():
ssub = Set(sub)
if len(sub) != len(ssub):
raise ValueError("repeated elements in {}".format(sub))
sub = ssub
try:
index_list = sorted(self._s.rank(x) for x in sub)
except (ValueError, IndexError):
raise ValueError("{} is not a subset of {}".format(
Set(sub), self._s))
n = self._s.cardinality()
r = sum(binomial(n, i) for i in range(len(index_list)))
return r + combination.rank(index_list, n)
def _b_power_k_r(self, k, r):
r"""
An expression involving Moebius inversion in the powersum generators.
For a positive value of ``k``, this expression is
.. MATH::
\sum_{j=0}^r (-1)^{r-j}k^j\binom{r,j} \left(
\frac{1}{k} \sum_{d|k} \mu(d/k) p_d \right)_k.
INPUT:
- ``k``, ``r`` -- positive integers
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st()
sage: st._b_power_k_r(1,1)
-p[] + p[1]
sage: st._b_power_k_r(2,2)
p[] + 4*p[1] + p[1, 1] - 4*p[2] - 2*p[2, 1] + p[2, 2]
sage: st._b_power_k_r(3,2)
p[] + 5*p[1] + p[1, 1] - 5*p[3] - 2*p[3, 1] + p[3, 3]
"""
p = self._p
return p.sum(
(-1)**(r - j) * k**j * binomial(r, j) *
p.prod(self._b_power_k(k) - i * p.one() for i in range(j))
for j in range(r + 1))
def log_gamma_binomial(p, gamma, n, M):
r"""
Return the list of coefficients in the power series
expansion (up to precision `M`) of `\binom{\log_p(z)/\log_p(\gamma)}{n}`
INPUT:
- ``p`` -- prime
- ``gamma`` -- topological generator, e.g. `1+p`
- ``n`` -- nonnegative integer
- ``M`` -- precision
OUTPUT:
The list of coefficients in the power series expansion of
`\binom{\log_p(z)/\log_p(\gamma)}{n}`
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import log_gamma_binomial
sage: log_gamma_binomial(5,1+5,2,4)
[0, -3/205, 651/84050, -223/42025]
sage: log_gamma_binomial(5,1+5,3,4)
[0, 2/205, -223/42025, 95228/25845375]
"""
S = PowerSeriesRing(QQ, 'z')
L = S([0] + [ZZ(-1)**j / j for j in range(1, M)]) # log_p(1+z)
loggam = L.O(M) / L(gamma - 1)
# log_{gamma}(1+z)= log_p(1+z)/log_p(gamma)
return binomial(loggam, n).list()
def cardinality(self):
r"""
Return the number of words in the shuffle product
of ``w1`` and ``w2``.
This is understood as a multiset cardinality, not as a
set cardinality; it does not count the distinct words only.
It is given by `\binom{l_1+l_2}{l_1}`, where `l_1` is the
length of ``w1`` and where `l_2` is the length of ``w2``.
EXAMPLES::
sage: from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2
sage: w, u = map(Words("abcd"), ["ab", "cd"])
sage: S = ShuffleProduct_w1w2(w,u)
sage: S.cardinality()
6
sage: w, u = map(Words("ab"), ["ab", "ab"])
sage: S = ShuffleProduct_w1w2(w,u)
sage: S.cardinality()
6
"""
return binomial(self._w1.length() + self._w2.length(),
self._w1.length())
def _induced_flags(self, n, tg, type_edges):
flag_counts = {}
flags = []
total = 0
for p in Tuples([0, 1], binomial(n, 2) - binomial(tg.n, 2)):
edges = list(type_edges)
c = 0
for i in range(tg.n + 1, n + 1):
for j in range(1, i):
if p[c] == 0:
edges.append((i, j))
else:
edges.append((j, i))
c += 1
ig = ThreeGraphFlag()
ig.n = n
ig.t = tg.n
for s in Combinations(range(1, n + 1), 3):
if self._variant:
if ((s[1], s[0]) in edges and (s[0], s[2]) in edges) or (
(s[2], s[0]) in edges and (s[0], s[1]) in edges):
ig.add_edge(s)
else:
if ((s[0], s[1]) in edges and (s[1], s[2]) in edges and (s[2], s[0]) in edges) or (
(s[0], s[2]) in edges and (s[2], s[1]) in edges and (s[1], s[0]) in edges):
ig.add_edge(s)
it = ig.induced_subgraph(range(1, tg.n + 1))
if tg.is_labelled_isomorphic(it):
ig.make_minimal_isomorph()
ghash = hash(ig)
if ghash in flag_counts:
flag_counts[ghash] += 1
else:
flags.append(ig)
flag_counts[ghash] = 1
total += 1
return [(f, flag_counts[hash(f)] / Integer(total)) for f in flags]
def unrank(self, r):
"""
EXAMPLES::
sage: c = Combinations([1,2,3])
sage: c.list() == list(map(c.unrank, range(c.cardinality())))
True
"""
k = 0
n = len(self.mset)
b = binomial(n, k)
while r >= b:
r -= b
k += 1
b = binomial(n, k)
return [self.mset[i] for i in from_rank(r, n, k)]
def f(partition):
n = 0
m = 1
for part in partition:
n += part
m *= q**binomial(part, 2)/q_factorial(part, q=q)
return t**n * m
def unrank(self, r):
"""
EXAMPLES::
sage: c = Combinations([1,2,3])
sage: c.list() == map(c.unrank, range(c.cardinality()))
True
"""
k = 0
n = len(self.mset)
b = binomial(n, k)
while r >= b:
r -= b
k += 1
b = binomial(n,k)
return [self.mset[i] for i in from_rank(r, n, k)]
def cardinality(self):
"""
Return the cardinality of ``self``.
EXAMPLES::
sage: IntegerVectors(3, 3, min_part=1).cardinality()
1
sage: IntegerVectors(5, 3, min_part=1).cardinality()
6
sage: IntegerVectors(13, 4, max_part=4).cardinality()
20
sage: IntegerVectors(k=4, max_part=3).cardinality()
256
sage: IntegerVectors(k=3, min_part=2, max_part=4).cardinality()
27
sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality()
16
"""
if self.k is None:
if self.n is None:
return PlusInfinity()
if ('max_length' not in self.constraints
and self.constraints.get('min_part', 0) <= 0):
return PlusInfinity()
elif ('max_part' in self.constraints
and self.constraints['max_part'] != PlusInfinity()):
if (self.n is None and len(self.constraints) == 2
and 'min_part' in self.constraints
and self.constraints['min_part'] >= 0):
num = self.constraints['max_part'] - self.constraints[
'min_part'] + 1
return Integer(num**self.k)
if len(self.constraints) == 1:
m = self.constraints['max_part']
if self.n is None:
return Integer((m + 1)**self.k)
if m >= self.n:
return Integer(binomial(self.n + self.k - 1, self.n))
# do by inclusion / exclusion on the number
# i of parts greater than m
return Integer(sum( (-1)**i * binomial(self.n+self.k-1-i*(m+1), self.k-1) \
* binomial(self.k,i) for i in range(self.n/(m+1)+1) ))
return ZZ.sum(ZZ.one() for x in self)
def cardinality(self):
"""
Return the size of combinations(set, k).
EXAMPLES::
sage: Combinations(range(16000), 5).cardinality()
8732673194560003200
"""
return binomial(len(self.mset), self.k)
def cardinality(self):
r"""
Returns the number of subwords of w of length k.
EXAMPLES::
sage: Subwords([1,2,3], 2).cardinality()
3
"""
return arith.binomial(Integer(len(self._w)), self._k)
def cardinality(self):
r"""
Returns the number of subwords of w of length k.
EXAMPLES::
sage: Subwords([1,2,3], 2).cardinality()
3
"""
return arith.binomial(Integer(len(self._w)), self._k)
def tdesign_params(t, v, k, L):
"""
Return the design's parameters: `(t, v, b, r , k, L)`. Note that `t` must be
given.
EXAMPLES::
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: from sage.combinat.designs.block_design import tdesign_params
sage: tdesign_params(2,7,3,1)
(2, 7, 7, 3, 3, 1)
"""
x = binomial(v, t)
y = binomial(k, t)
b = divmod(L * x, y)[0]
x = binomial(v-1, t-1)
y = binomial(k-1, t-1)
r = integer_floor(L * x/y)
return (t, v, b, r, k, L)
def cardinality(self):
"""
Return the cardinality of ``self``.
EXAMPLES::
sage: IntegerVectors(3, 3, min_part=1).cardinality()
1
sage: IntegerVectors(5, 3, min_part=1).cardinality()
6
sage: IntegerVectors(13, 4, max_part=4).cardinality()
20
sage: IntegerVectors(k=4, max_part=3).cardinality()
256
sage: IntegerVectors(k=3, min_part=2, max_part=4).cardinality()
27
sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality()
16
"""
if self.k is None:
if self.n is None:
return PlusInfinity()
if ('max_length' not in self.constraints
and self.constraints.get('min_part', 0) <= 0):
return PlusInfinity()
elif ('max_part' in self.constraints
and self.constraints['max_part'] != PlusInfinity()):
if (self.n is None and len(self.constraints) == 2
and 'min_part' in self.constraints
and self.constraints['min_part'] >= 0):
num = self.constraints['max_part'] - self.constraints['min_part'] + 1
return Integer(num ** self.k)
if len(self.constraints) == 1:
m = self.constraints['max_part']
if self.n is None:
return Integer((m + 1) ** self.k)
if m >= self.n:
return Integer(binomial(self.n + self.k - 1, self.n))
# do by inclusion / exclusion on the number
# i of parts greater than m
return Integer(sum( (-1)**i * binomial(self.n+self.k-1-i*(m+1), self.k-1) \
* binomial(self.k,i) for i in range(self.n/(m+1)+1) ))
return ZZ.sum(ZZ.one() for x in self)
def tdesign_params(t, v, k, L):
"""
Return the design's parameters: `(t, v, b, r , k, L)`. Note that `t` must be
given.
EXAMPLES::
sage: BD = BlockDesign(7,[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: from sage.combinat.designs.block_design import tdesign_params
sage: tdesign_params(2,7,3,1)
(2, 7, 7, 3, 3, 1)
"""
x = binomial(v, t)
y = binomial(k, t)
b = divmod(L * x, y)[0]
x = binomial(v - 1, t - 1)
y = binomial(k - 1, t - 1)
r = integer_floor(L * x / y)
return (t, v, b, r, k, L)
def by_taylor_expansion(self, fs, k, is_integral=False):
r"""
We combine the theta decomposition and the heat operator as in [Sko].
This yields a bijections of Jacobi forms of weight `k` and
`M_k \times S_{k+2} \times .. \times S_{k+2m}`.
"""
## we introduce an abbreviations
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1:
raise ValueError(
"fs must be a list of m + 1 elliptic modular forms or their fourier expansion"
)
qexp_prec = self._qexp_precision()
if qexp_prec is None: # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs):
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
if self.__precision.jacobi_index() == 1:
return self._by_taylor_expansion_m1(f_divs, k, is_integral)
for i in xrange(self.__precision.jacobi_index() + 1):
for j in xrange(1, self.__precision.jacobi_index() - i + 1):
f_divs[(i, j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1):
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append(
sum(f_divs[(j, i - j)] *
(4 * self.__precision.jacobi_index())**i * binomial(i, j) /
2**i #2**(self.__precision.jacobi_index() - i + 1)
* prod(2 * (i - l) + 1
for l in xrange(1, i)) / factorial(i + k + j - 1) *
factorial(2 * self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1)))
phi_coeffs = dict()
for r in xrange(self.__precision.jacobi_index() + 1):
series = sum(map(operator.mul, self._theta_factors()[r], phi_divs))
series = self._eta_factor() * series
for n in xrange(qexp_prec):
phi_coeffs[(n, r)] = series[n]
return phi_coeffs
def cardinality(self):
"""
Returns the number of multichoices of k things from a list of n
things.
EXAMPLES::
sage: MultichooseNK(3,2).cardinality()
6
"""
n, k = self._n, self._k
return binomial(n + k - 1, k)
def cardinality(self):
"""
Returns the number of multichoices of k things from a list of n
things.
EXAMPLES::
sage: MultichooseNK(3,2).cardinality()
6
"""
n,k = self._n, self._k
return binomial(n+k-1,k)
def cardinality(self):
r"""
Return the cardinality of ``self``.
EXAMPLES::
sage: from sage.combinat.blob_algebra import BlobDiagrams
sage: BD4 = BlobDiagrams(4)
sage: BD4.cardinality()
70
"""
return binomial(2 * self._n, self._n)
def _induced_flags(self, n, tg, type_edges):
flag_counts = {}
flags = []
total = 0
for p in Tuples([0, 1], binomial(n, 2) - binomial(tg.n, 2)):
edges = list(type_edges)
c = 0
for i in range(tg.n + 1, n + 1):
for j in range(1, i):
if p[c] == 1:
edges.append((j, i))
c += 1
ig = ThreeGraphFlag()
ig.n = n
ig.t = tg.n
for s in Combinations(range(1, n + 1), 3):
ind_edges = [e for e in edges if e[0] in s and e[1] in s]
if len(ind_edges) == 1 or len(ind_edges) == 3:
ig.add_edge(s)
it = ig.induced_subgraph(range(1, tg.n + 1))
if tg.is_labelled_isomorphic(it):
ig.make_minimal_isomorph()
ghash = hash(ig)
if ghash in flag_counts:
flag_counts[ghash] += 1
else:
flags.append(ig)
flag_counts[ghash] = 1
total += 1
return [(f, flag_counts[hash(f)] / Integer(total)) for f in flags]
def volume_hamming(n, q, r):
r"""
Return the number of elements in a Hamming ball.
Return the number of elements in a Hamming ball of radius `r` in
`\GF{q}^n`.
EXAMPLES::
sage: codes.bounds.volume_hamming(10,2,3)
176
"""
return sum([binomial(n, i) * (q - 1)**i for i in range(r + 1)])
def by_taylor_expansion(self, fs, k, is_integral=False) :
r"""
We combine the theta decomposition and the heat operator as in [Sko].
This yields a bijections of Jacobi forms of weight `k` and
`M_k \times S_{k+2} \times .. \times S_{k+2m}`.
"""
## we introduce an abbreviations
if is_integral :
PS = self.integral_power_series_ring()
else :
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1 :
raise ValueError("fs must be a list of m + 1 elliptic modular forms or their fourier expansion")
qexp_prec = self._qexp_precision()
if qexp_prec is None : # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs) :
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
if self.__precision.jacobi_index() == 1 :
return self._by_taylor_expansion_m1(f_divs, k, is_integral)
for i in xrange(self.__precision.jacobi_index() + 1) :
for j in xrange(1, self.__precision.jacobi_index() - i + 1) :
f_divs[(i,j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1) :
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append( sum( f_divs[(j, i - j)] * (4 * self.__precision.jacobi_index())**i
* binomial(i,j) / 2**i#2**(self.__precision.jacobi_index() - i + 1)
* prod(2*(i - l) + 1 for l in xrange(1, i))
/ factorial(i + k + j - 1)
* factorial(2*self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1) ) )
phi_coeffs = dict()
for r in xrange(self.__precision.jacobi_index() + 1) :
series = sum( map(operator.mul, self._theta_factors()[r], phi_divs) )
series = self._eta_factor() * series
for n in xrange(qexp_prec) :
phi_coeffs[(n, r)] = series[n]
return phi_coeffs
def rank(self, x):
"""
EXAMPLES::
sage: c = Combinations([1,2,3])
sage: range(c.cardinality()) == map(c.rank, c)
True
"""
x = [self.mset.index(_) for _ in x]
r = 0
n = len(self.mset)
for i in range(len(x)):
r += binomial(n, i)
r += rank(x, n)
return r
def cardinality(self):
"""
EXAMPLES::
sage: IntegerVectors(3,3, min_part=1).cardinality()
1
sage: IntegerVectors(5,3, min_part=1).cardinality()
6
sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality()
16
"""
if not self._constraints:
if self.k < 0:
return +infinity
if self.n >= 0:
return binomial(self.n + self.k - 1, self.n)
else:
return 0
else:
if len(
self._constraints
) == 1 and 'max_part' in self._constraints and self._constraints[
'max_part'] != infinity:
m = self._constraints['max_part']
if m >= self.n:
return binomial(self.n + self.k - 1, self.n)
else: #do by inclusion / exclusion on the number
#i of parts greater than m
return sum([
(-1)**i * binomial(self.n + self.k - 1 - i *
(m + 1), self.k - 1) *
binomial(self.k, i)
for i in range(self.n // (m + 1) + 1)
])
else:
return super(IntegerVectors_nkconstraints, self).cardinality()
def rank(self, x):
"""
EXAMPLES::
sage: c = Combinations([1,2,3])
sage: list(range(c.cardinality())) == list(map(c.rank, c))
True
"""
x = [self.mset.index(_) for _ in x]
r = 0
n = len(self.mset)
for i in range(len(x)):
r += binomial(n, i)
r += rank(x, n)
return r
def s(f,k):
"""Given f monic of degree n with distinct roots, returns the monic
polynomial of degree binomial(n,k) whose roots are all products of
k distinct roots of f.
"""
n = f.degree()
x = f.variables()[0]
if k==0:
return x-1
if k==1:
return f
c = (-1)**n * f(0)
if k==n:
return x-c
if k>n-k: # use s(f,n-k)
g = s(f,n-k)
from sage.arith.all import binomial
return ((-x)**binomial(n,k) * g(c/x) / c**binomial(n-1,k)).numerator()
if k==2:
return (star(f,f)//r(f,2)).nth_root(2)
if k==3:
f2 = s(f,2)
return (star(f2,f)*r(f,3) // star(r(f,2),f)).nth_root(3)
fkn = fkd = 1
for j in range(1,k+1):
g = star(r(f,j), s(f,k-j))
if j%2:
fkn *= g
else:
fkd *= g
fk = fkn//fkd
assert fk*fkd==fkn
return fk.nth_root(k)
def cardinality(self):
"""
Returns the number of multichoices of k things from a list of n
things.
EXAMPLES::
sage: MultichooseNK(3,2).cardinality()
doctest:...: DeprecationWarning: MultichooseNK should be
replaced by itertools.combinations_with_replacement
See http://trac.sagemath.org/16473 for details.
6
"""
n, k = self._n, self._k
return binomial(n + k - 1, k)
def _deflated(cls, self, a, b, z):
"""
Private helper to return list of deflated terms.
EXAMPLES::
sage: x = hypergeometric([5], [4], 3)
sage: y = x.deflated()
sage: y
7/4*hypergeometric((), (), 3)
sage: x.n(); y.n()
35.1496896155784
35.1496896155784
"""
new = self.eliminate_parameters()
aa = new.operands()[0].operands()
bb = new.operands()[1].operands()
for i, aaa in enumerate(aa):
for j, bbb in enumerate(bb):
m = aaa - bbb
if m in ZZ and m > 0:
aaaa = aa[:i] + aa[i + 1:]
bbbb = bb[:j] + bb[j + 1:]
terms = []
for k in xrange(m + 1):
# TODO: could rewrite prefactors as recurrence
term = binomial(m, k)
for c in aaaa:
term *= rising_factorial(c, k)
for c in bbbb:
term /= rising_factorial(c, k)
term *= z ** k
term /= rising_factorial(aaa - m, k)
F = hypergeometric([c + k for c in aaaa],
[c + k for c in bbbb], z)
unique = []
counts = []
for c, f in F._deflated():
if f in unique:
counts[unique.index(f)] += c
else:
unique.append(f)
counts.append(c)
Fterms = zip(counts, unique)
terms += [(term * termG, G) for (termG, G) in
Fterms]
return terms
return ((1, new),)
def cardinality(self):
r"""
Return the number of shuffles of `l_1` and `l_2`, respectively of lengths `m` and
`n`, which is `\binom{m+n}{n}`.
TESTS::
sage: from sage.combinat.shuffle import ShuffleProduct
sage: ShuffleProduct([3,1,2], [4,2,1,3]).cardinality()
35
sage: ShuffleProduct([3,1,2,5,6,4], [4,2,1,3]).cardinality() == binomial(10,4)
True
"""
ll1 = len(self._l1)
ll2 = len(self._l2)
return binomial(ll1 + ll2, ll1)
def linial(self, n, K=QQ, names=None):
r"""
Return the linial hyperplane arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default: `\QQ`)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The linial hyperplane arrangement is the set of hyperplanes
`\{x_i - x_j = 1 : 1\leq i < j \leq n\}`.
EXAMPLES::
sage: a = hyperplane_arrangements.linial(4); a
Arrangement of 6 hyperplanes of dimension 4 and rank 3
sage: a.characteristic_polynomial()
x^4 - 6*x^3 + 15*x^2 - 14*x
TESTS::
sage: h = hyperplane_arrangements.linial(5)
sage: h.characteristic_polynomial()
x^5 - 10*x^4 + 45*x^3 - 100*x^2 + 90*x
sage: h.characteristic_polynomial.clear_cache() # long time
sage: h.characteristic_polynomial() # long time
x^5 - 10*x^4 + 45*x^3 - 100*x^2 + 90*x
"""
H = make_parent(K, n, names)
x = H.gens()
hyperplanes = []
for i in range(n):
for j in range(i+1, n):
hyperplanes.append(x[i] - x[j] - 1)
A = H(*hyperplanes)
x = polygen(QQ, 'x')
charpoly = x * sum(binomial(n, k)*(x - k)**(n - 1) for k in range(n + 1)) / 2**n
A.characteristic_polynomial.set_cache(charpoly)
return A
def _comb_largest(a,b,x):
r"""
Returns the largest `w < a` such that `binomial(w,b) <= x`.
EXAMPLES::
sage: from sage.combinat.combination import _comb_largest
sage: _comb_largest(6,3,10)
5
sage: _comb_largest(6,3,5)
4
"""
w = a - 1
while binomial(w,b) > x:
w -= 1
return w
def cardinality(self):
r"""
Return the number of elements in ``self``.
The number of signed compositions of `n` is equal to
.. MATH::
\sum_{i=1}^{n+1} \binom{n-1}{i-1} 2^i
TESTS::
sage: SC4 = SignedCompositions(4)
sage: SC4.cardinality() == len(SC4.list())
True
sage: SignedCompositions(3).cardinality()
18
"""
return sum([binomial(self.n-1, i-1)*2**(i) for i in range(1, self.n+1)])
def generating_series(self, weight = None):
r"""
Returns a length generating series for the elements of ``self``
EXAMPLES::
sage: W = WeylGroup(["A", 3, 1])
sage: W.pieri_factors().cardinality()
15
sage: W.pieri_factors().generating_series()
4*z^3 + 6*z^2 + 4*z + 1
"""
if weight is None:
weight = self.default_weight()
l_min = len(self._min_support)
l_max = len(self._max_support)
return sum(binomial(l_max-l_min, l-l_min) * weight(l)
for l in range(self._min_length, self._max_length+1))
def zero_eigenvectors(self, tg, flags):
rows = set()
for p in Tuples([0, 1], binomial(tg.n, 2)):
edges = []
c = 0
for i in range(1, tg.n + 1):
for j in range(1, i):
if p[c] == 1:
edges.append((j, i))
c += 1
graphs = self._induced_flags(flags[0].n, tg, edges)
row = [0 for f in flags]
for pair in graphs:
g, den = pair
for i in range(len(flags)):
if g.is_labelled_isomorphic(flags[i]):
row[i] = den
break
rows.add(tuple(row))
return matrix_of_independent_rows(self._field, list(rows), len(flags))
def unrank(self, r):
"""
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)
"""
r = Integer(r)
if r >= self.cardinality() or r < 0:
raise IndexError("index out of range")
k = ZZ_0
n = self._s.cardinality()
binom = ZZ.one()
while r >= binom:
r -= binom
k += 1
binom = binomial(n,k)
C = combination.from_rank(r, n, k)
return self.element_class(sorted([self._s.unrank(i) for i in C]))
def cardinality(self):
"""
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
"""
if self._k > self._s.cardinality():
return ZZ_0
return binomial(self._s.cardinality(), self._k)
def saturation(A, proof=True, p=0, max_dets=5):
"""
Compute a saturation matrix of A.
INPUT:
- A -- a matrix over ZZ
- proof -- bool (default: True)
- p -- int (default: 0); if not 0 only guarantees that output is
p-saturated
- max_dets -- int (default: 4) max number of dets of submatrices to
compute.
OUTPUT:
matrix -- saturation of the matrix A.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_saturation import saturation
sage: A = matrix(ZZ, 2, 2, [3,2,3,4]); B = matrix(ZZ, 2,3,[1,2,3,4,5,6]); C = A*B
sage: C
[11 16 21]
[19 26 33]
sage: C.index_in_saturation()
18
sage: S = saturation(C); S
[11 16 21]
[-2 -3 -4]
sage: S.index_in_saturation()
1
sage: saturation(C, proof=False)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2, max_dets=1)
[11 16 21]
[-2 -3 -4]
"""
# Find a submatrix of full rank and instead saturate that matrix.
r = A.rank()
if A.is_square() and r == A.nrows():
return identity_matrix(ZZ, r)
if A.nrows() > r:
P = []
while len(P) < r:
P = matrix_integer_dense_hnf.probable_pivot_rows(A)
A = A.matrix_from_rows(P)
# Factor out all common factors from all rows, just in case.
A = copy(A)
A._factor_out_common_factors_from_each_row()
if A.nrows() <= 1:
return A
A, zero_cols = A._delete_zero_columns()
if max_dets > 0:
# Take the GCD of at most num_dets randomly chosen determinants.
nr = A.nrows(); nc = A.ncols()
d = 0
trials = min(binomial(nc, nr), max_dets)
already_tried = []
while len(already_tried) < trials:
v = random_sublist_of_size(nc, nr)
tm = verbose('saturation -- checking det condition on submatrix')
d = gcd(d, A.matrix_from_columns(v).determinant(proof=proof))
verbose('saturation -- got det down to %s'%d, tm)
if gcd(d, p) == 1:
return A._insert_zero_columns(zero_cols)
already_tried.append(v)
if gcd(d, p) == 1:
# already p-saturated
return A._insert_zero_columns(zero_cols)
# Factor and p-saturate at each p.
# This is not a good algorithm, because all the HNF's in it are really slow!
#
#tm = verbose('factoring gcd %s of determinants'%d)
#limit = 2**31-1
#F = d.factor(limit = limit)
#D = [p for p, e in F if p <= limit]
#B = [n for n, e in F if n > limit] # all big factors -- there will only be at most one
#assert len(B) <= 1
#C = B[0]
#for p in D:
# A = p_saturation(A, p=p, proof=proof)
# This is a really simple but powerful algorithm.
# FACT: If A is a matrix of full rank, then hnf(transpose(A))^(-1)*A is a saturation of A.
# To make this practical we use solve_system_with_difficult_last_row, since the
# last column of HNF's are typically the only really big ones.
B = A.transpose().hermite_form(include_zero_rows=False, proof=proof)
B = B.transpose()
# Now compute B^(-1) * A
C = solve_system_with_difficult_last_row(B, A)
return C.change_ring(ZZ)._insert_zero_columns(zero_cols)