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 xrange(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.rings.arith 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 xrange(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 = 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):
"""
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.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 len(self.list())
def parameters(self, t=None):
"""
Returns `(t,v,k,lambda)`. Does not check if the input is a block
design.
INPUT:
- ``t`` -- `t` such that the design is a `t`-design.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
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]], name="FanoPlane")
sage: BD.parameters(t=2)
(2, 7, 3, 1)
sage: BD.parameters(t=3)
(3, 7, 3, 0)
"""
if t is None:
from sage.misc.superseded import deprecation
deprecation(
15664, "the 't' argument will become mandatory soon. 2" +
" is used when none is provided.")
t = 2
v = len(self.points())
blks = self.blocks()
k = len(blks[int(0)])
b = len(blks)
#A = self.incidence_matrix()
#r = sum(A.rows()[0])
lmbda = int(b / (binomial(v, t) / binomial(k, t)))
return (t, v, k, lmbda)
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 xrange(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.rings.arith 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 xrange(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.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 len(self.list())
def parameters(self, t=None):
"""
Returns `(t,v,k,lambda)`. Does not check if the input is a block
design.
INPUT:
- ``t`` -- `t` such that the design is a `t`-design.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
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]], name="FanoPlane")
sage: BD.parameters(t=2)
(2, 7, 3, 1)
sage: BD.parameters(t=3)
(3, 7, 3, 0)
"""
if t is None:
from sage.misc.superseded import deprecation
deprecation(15664, "the 't' argument will become mandatory soon. 2"+
" is used when none is provided.")
t = 2
v = len(self.points())
blks = self.blocks()
k = len(blks[int(0)])
b = len(blks)
#A = self.incidence_matrix()
#r = sum(A.rows()[0])
lmbda = int(b/(binomial(v, t)/binomial(k, t)))
return (t, v, k, lmbda)
def _basic_integral(self, a, j, twist=None):
r"""
Computes the integral
.. MATH::
\int_{a+p\ZZ_p}(z-\omega(a))^jd\mu_\chi.
If ``twist`` is ``None``, `\\chi` is the trivial character. Otherwise, ``twist`` can be a primitive quadratic character of conductor prime to `p`.
"""
#is this the negative of what we want?
#if Phis is fixed for this p-adic L-function, we should make this method cached
p = self._Phis.parent().prime()
if twist is None:
pass
elif twist in ZZ:
twist = kronecker_character(twist)
if twist.is_trivial():
twist = None
else:
D = twist.level()
assert(D.gcd(p) == 1)
else:
if twist.is_trivial():
twist = None
else:
assert((twist**2).is_trivial())
twist = twist.primitive_character()
D = twist.level()
assert(D.gcd(p) == 1)
onDa = self._on_Da(a, twist)#self._Phis(Da)
aminusat = a - self._Phis.parent().base_ring().base_ring().teichmuller(a)
#aminusat = a - self._coefficient_ring.base_ring().teichmuller(a)
try:
ap = self._ap
except AttributeError:
self._ap = self._Phis.Tq_eigenvalue(p) #catch exception if not eigensymbol
ap = self._ap
if not twist is None:
ap *= twist(p)
if j == 0:
return (~ap) * onDa.moment(0)
if a == 1:
#aminusat is 0, so only the j=r term is non-zero
return (~ap) * (p ** j) * onDa.moment(j)
#print "j =", j, "a = ", a
ans = onDa.moment(0) * (aminusat ** j)
#ans = onDa.moment(0)
#print "\tr =", 0, " ans =", ans
for r in range(1, j+1):
if r == j:
ans += binomial(j, r) * (p ** r) * onDa.moment(r)
else:
ans += binomial(j, r) * (aminusat ** (j - r)) * (p ** r) * onDa.moment(r)
#print "\tr =", r, " ans =", ans
#print " "
return (~ap) * ans
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 from_rank(r, n, k):
"""
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: http://en.wikipedia.org/wiki/Combinadic
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.from_rank(0,3,0)
()
sage: choose_nk.from_rank(0,3,1)
(0,)
sage: choose_nk.from_rank(1,3,1)
(1,)
sage: choose_nk.from_rank(2,3,1)
(2,)
sage: choose_nk.from_rank(0,3,2)
(0, 1)
sage: choose_nk.from_rank(1,3,2)
(0, 2)
sage: choose_nk.from_rank(2,3,2)
(1, 2)
sage: choose_nk.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 from_rank(r, n, k):
"""
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: http://en.wikipedia.org/wiki/Combinadic
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.from_rank(0,3,0)
()
sage: choose_nk.from_rank(0,3,1)
(0,)
sage: choose_nk.from_rank(1,3,1)
(1,)
sage: choose_nk.from_rank(2,3,1)
(2,)
sage: choose_nk.from_rank(0,3,2)
(0, 1)
sage: choose_nk.from_rank(1,3,2)
(0, 2)
sage: choose_nk.from_rank(2,3,2)
(1, 2)
sage: choose_nk.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 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 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 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 __getitem__(self, key):
"""
EXAMPLES::
sage: a, b, c, q, z = var('a b c q z')
sage: bhs = BasicHypergeometricSeries([a, b], [c], q, z)
sage: for i in range(4): print bhs[i]
1
(b - 1)*(a - 1)*z/((q - 1)*(c - 1))
(b - 1)*(a - 1)*(b*q - 1)*(a*q - 1)*z^2/((q - 1)*(c - 1)*(q^2 - 1)*(c*q - 1))
(b - 1)*(a - 1)*(b*q - 1)*(a*q - 1)*(b*q^2 - 1)*(a*q^2 - 1)*z^3/((q - 1)*(c - 1)*(q^2 - 1)*(c*q - 1)*(q^3 - 1)*(c*q^2 - 1))
"""
if key >= 0:
j, k = len(self.list_a), len(self.list_b)
nominator = qPochhammerSymbol(self.list_a, self.q, key).evaluate()
if nominator == 0:
return 0
denominator = (
qPochhammerSymbol(self.list_b, self.q, key).evaluate()
* qPochhammerSymbol(self.q, self.q, key).evaluate()
)
return nominator / denominator * ((-1) ** key * self.q ** (binomial(key, 2))) ** (1 + k - j) * self.z ** key
else:
return 0
def loggam_binom(p,gam,z,n,M):
r"""
Returns the list of coefficients in the power series
expansion (up to precision `M`) of `{\log_p(z)/\log_p(\gamma) \choose n}`
INPUT:
- ``p`` -- prime
- ``gam`` -- topological generator e.g., `1+p`
- ``z`` -- variable
- ``n`` -- nonnegative integer
- ``M`` -- precision
OUTPUT:
The list of coefficients in the power series expansion of
`{\log_p(z)/\log_p(\gamma) \choose n}`
EXAMPLES:
sage: R.<z> = QQ['z']
sage: loggam_binom(5,1+5,z,2,4)
[0, -3/205, 651/84050, -223/42025]
sage: loggam_binom(5,1+5,z,3,4)
[0, 2/205, -223/42025, 95228/25845375]
"""
L = logp(p,z,M)
logpgam = L.substitute(z = (gam-1)) #log base p of gamma
loggam = L/logpgam #log base gamma
return z.parent()(binomial(loggam,n)).truncate(M).list()
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 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])
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 xrange(len(index_list)))
return r + choose_nk.rank(index_list, n)
def log_gamma_binomial(p,gamma,z,n,M):
r"""
Returns the list of coefficients in the power series
expansion (up to precision `M`) of `{\log_p(z)/\log_p(\gamma) \choose 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
`{\log_p(z)/\log_p(\gamma) \choose 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 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 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 basic_integral(Phi,a,j,ap,D):
"""
Returns `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``Phi`` -- overconvergnt `U_p`-eigensymbol
- ``a`` -- integer in [0..p-1]
- ``j`` -- positive integer
- ``ap`` -- Hecke eigenvalue?
- ``D`` -- conductor of the quadratic twist `\chi`
OUTPUT:
`\int_{a+pZ_p} (z-{a})^j d\Phi(0-\infty)`
EXAMPLES:
"""
M = Phi.num_moments()
p = Phi.p()
ap = ap*kronecker(D,p)
ans = 0
for r in range(j+1):
ans = ans+binomial(j,r)*((a-teich(a,p,M))**(j-r))*(p**r)*phi_on_Da(Phi,a,D).moment(r)
return ans/ap
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 unrank(self, r):
"""
Returns 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 choose_nk.from_rank(r, n, k)])
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])
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 xrange(len(index_list)))
return r + choose_nk.rank(index_list,n)
def Krawtchouk(n,q,l,i):
"""
Compute ``K^{n,q}_l(i)``, the Krawtchouk polynomial:
see :wikipedia:`Kravchuk_polynomials`.
It is given by
.. math::
K^{n,q}_l(i)=\sum_{j=0}^l (-1)^j(q-1)^{(l-j)}{i \choose j}{n-i \choose l-j}
EXAMPLES::
sage: Krawtchouk(24,2,5,4)
2224
sage: Krawtchouk(12300,4,5,6)
567785569973042442072
"""
from sage.rings.arith import binomial
# Use the expression in equation (55) of MacWilliams & Sloane, pg 151
# We write jth term = some_factor * (j-1)th term
kraw = jth_term = (q-1)**l * binomial(n, l) # j=0
for j in range(1,l+1):
jth_term *= -q*(l-j+1)*(i-j+1)/((q-1)*j*(n-j+1))
kraw += jth_term
return kraw
def unrank(self, r):
"""
Returns 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 choose_nk.from_rank(r, n, k)])
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 not in range(len(self.s) + 1):
return 0
else:
return binomial(len(self.s), self.k)
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 not in range(len(self.s)+1):
return 0
else:
return binomial(len(self.s),self.k)
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 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 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 rank(comb, n):
"""
Returns the rank of comb in the subsets of range(n) of size k.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See: http://en.wikipedia.org/wiki/Combinadic
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.rank([], 3)
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
"""
k = len(comb)
if k > n:
raise ValueError, "len(comb) must be <= n"
#Generate the combinadic from the
#combination
w = [0]*k
for i in range(k):
w[i] = (n-1) - comb[i]
#Calculate the integer that is the dual of
#the lexicographic index of the combination
r = k
t = 0
for i in range(k):
t += binomial(w[i],r)
r -= 1
return binomial(n,k)-t-1
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 rank(comb, n):
"""
Returns the rank of comb in the subsets of range(n) of size k.
The algorithm used is based on combinadics and James McCaffrey's
MSDN article. See: http://en.wikipedia.org/wiki/Combinadic
EXAMPLES::
sage: import sage.combinat.choose_nk as choose_nk
sage: choose_nk.rank([], 3)
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
"""
k = len(comb)
if k > n:
raise ValueError, "len(comb) must be <= n"
#Generate the combinadic from the
#combination
w = [0] * k
for i in range(k):
w[i] = (n - 1) - comb[i]
#Calculate the integer that is the dual of
#the lexicographic index of the combination
r = k
t = 0
for i in range(k):
t += binomial(w[i], r)
r -= 1
return binomial(n, k) - t - 1
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 map(lambda i: self.mset[i], from_rank(r, n, k))
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 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 = designs.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):
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 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):
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 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 specialize(self):
"""specializes to weight k -- i.e. projects to Sym^k"""
k=self.weight
if k==0:
# R.<X,Y>=PolynomialRing(QQ,2)
R = PolynomialRing(QQ,('X','Y'))
X,Y = R.gens()
P=0
for j in range(0,k+1):
P=P+binomial(k,j)*((-1)**j)*self.moment(j)*(X**j)*(Y**(k-j))
return symk(k,P)
def dcoeff(l,m,N):
"""Returns the 'D' coefficient for partitions l, m having exactly N parts. See
\cite{Fulton:Intersection_Theory}"""
# make padded copies of l, m
l1 = copy(l)
m1 = copy(m)
l1.extend([0]*(N-len(l)))
m1.extend([0]*(N-len(m)))
# for the defining matrix
M = Matrix( [ [ binomial(l1[i]+N-(i+1), m1[j]+N-(j+1)) for j in range(N) ] for i in range(N) ])
return det(M)
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):
"""
Returns the number of choices of set partitions of range(n) into a
set of size k and a set of size n-k.
EXAMPLES::
sage: from sage.combinat.split_nk import SplitNK
sage: SplitNK(5,2).cardinality()
10
"""
return binomial(self._n,self._k)
def cardinality(self):
"""
Returns the number of choices of set partitions of range(n) into a
set of size k and a set of size n-k.
EXAMPLES::
sage: from sage.combinat.split_nk import SplitNK
sage: SplitNK(5,2).cardinality()
10
"""
return binomial(self._n,self._k)
def parameters(self, t=2):
"""
Returns (t,v,k,lambda). Does not check if the input is a block
design. Uses t=2 by default.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
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]], name="FanoPlane")
sage: BD.parameters()
(2, 7, 3, 1)
sage: BD.parameters(t=3)
(3, 7, 3, 0)
"""
v = len(self.points())
blks = self.blocks()
k = len(blks[int(0)])
b = len(blks)
#A = self.incidence_matrix()
#r = sum(A.rows()[0])
lmbda = int(b/(binomial(v,t)/binomial(k,t)))
return (t,v,k,lmbda)
def specialize(self):
"""specializes to weight k -- i.e. projects to Sym^k -- NO CHARACTER HERE!!"""
assert 0==1, "didn't program character here yet"
k=self.weight
if k==0:
return symk(0,self.moments[0])
else:
# R.<X,Y>=PolynomialRing(QQ,2)
R = PolynomialRing(QQ,('X','Y'))
P=0
for j in range(0,k+1):
P=P+binomial(k,j)*((-1)**j)*self.moments[j]*(X**j)*(Y**(k-j))
return symk(k,P)
def cardinality(self):
"""
Returns the number of choices of k things from a list of n things.
EXAMPLES::
sage: from sage.combinat.choose_nk import ChooseNK
sage: ChooseNK(3,2).cardinality()
3
sage: ChooseNK(5,2).cardinality()
10
"""
return binomial(self._n, self._k)
def cardinality(self):
"""
Returns the number of choices of k things from a list of n things.
EXAMPLES::
sage: from sage.combinat.choose_nk import ChooseNK
sage: ChooseNK(3,2).cardinality()
3
sage: ChooseNK(5,2).cardinality()
10
"""
return binomial(self._n, self._k)
def parameters(self, t=2):
"""
Returns (t,v,k,lambda). Does not check if the input is a block
design. Uses t=2 by default.
EXAMPLES::
sage: from sage.combinat.designs.block_design import BlockDesign
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]], name="FanoPlane")
sage: BD.parameters()
(2, 7, 3, 1)
sage: BD.parameters(t=3)
(3, 7, 3, 0)
"""
v = len(self.points())
blks = self.blocks()
k = len(blks[int(0)])
b = len(blks)
#A = self.incidence_matrix()
#r = sum(A.rows()[0])
lmbda = int(b/(binomial(v,t)/binomial(k,t)))
return (t,v,k,lmbda)
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 _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 _comb_largest(a, b, x):
"""
Returns the largest w < a such that binomial(w,b) <= x.
EXAMPLES::
sage: from sage.combinat.choose_nk 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 _check_pbd(B, v, S):
r"""
Checks that ``B`` is a PBD on `v` points with given block sizes.
INPUT:
- ``bibd`` -- a list of blocks
- ``v`` (integer) -- number of points
- ``S`` -- list of integers
EXAMPLE::
sage: designs.BalancedIncompleteBlockDesign(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 11], [0, 5, 7, 10],
[0, 13, 26, 39], [0, 14, 28, 38], [0, 15, 25, 27],
[0, 16, 32, 35], [0, 17, 34, 37], [0, 18, 33, 36],
...
"""
from itertools import combinations
from sage.graphs.graph import Graph
if not all(len(X) in S for X in B):
raise RuntimeError(
"This is not a nice honest PBD from the good old days !")
g = Graph()
m = 0
for X in B:
g.add_edges(list(combinations(X, 2)))
if g.size() != m + binomial(len(X), 2):
raise RuntimeError(
"This is not a nice honest PBD from the good old days !")
m = g.size()
if not (g.is_clique() and g.vertices() == range(v)):
raise RuntimeError(
"This is not a nice honest PBD from the good old days !")
return B