INPUT:
- ``R`` -- a ring.
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: Hom = E.point_homset(); Hom
Abelian group of points on Elliptic Curve defined
by y^2 + y = x^3 - x over Rational Field
sage: Hom.base_ring()
Integer Ring
sage: Hom.base_extend(QQ)
Traceback (most recent call last):
...
NotImplementedError: Abelian variety point sets are not
implemented as modules over rings other than ZZ
"""
if R is not ZZ:
raise NotImplementedError(
'Abelian variety point sets are not '
'implemented as modules over rings other than ZZ')
return self
from sage.misc.persist import register_unpickle_override
register_unpickle_override(
'sage.schemes.generic.homset',
'SchemeHomsetModule_abelian_variety_coordinates_field',
SchemeHomset_points_abelian_variety_field)
- an element of the base ring
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p([3,3]).eval_at_permutation_roots([6])
0
sage: p([3,3]).eval_at_permutation_roots([3])
9
sage: p([3,3]).eval_at_permutation_roots([1])
1
sage: p([3,3]).eval_at_permutation_roots([3,3])
36
sage: p([3,3]).eval_at_permutation_roots([1,1,1,1,1])
25
sage: (p[1]+p[2]+p[3]).eval_at_permutation_roots([3,2])
5
"""
p = self.parent()
R = self.base_ring()
on_basis = lambda lam: R.prod(
p.eval_at_permutation_roots_on_generators(k, rho) for k in lam)
return p._apply_module_morphism(self, on_basis, R)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.powersum',
'SymmetricFunctionAlgebraElement_power',
SymmetricFunctionAlgebra_power.Element)
sage: SC4 = SignedCompositions(4)
sage: SC4.cardinality() == len(SC4.list())
True
sage: SignedCompositions(3).cardinality()
18
"""
return ZZ.sum(
binomial(self.n - 1, i - 1) * 2**i for i in range(1, self.n + 1))
def __iter__(self):
"""
TESTS::
sage: SignedCompositions(0).list() #indirect doctest
[[]]
sage: SignedCompositions(1).list() #indirect doctest
[[1], [-1]]
sage: SignedCompositions(2).list() #indirect doctest
[[1, 1], [1, -1], [-1, 1], [-1, -1], [2], [-2]]
"""
for comp in Compositions_n.__iter__(self):
l = len(comp)
for sign in itertools.product([1, -1], repeat=l):
yield [sign[i] * comp[i] for i in range(l)]
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.composition_signed',
'SignedCompositions_n', SignedCompositions)
sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,4); m
Manin Symbol List of weight 4 for Gamma1(4) with character [-1]
sage: [m.normalize(s.tuple()) for s in m.manin_symbol_list()]
[((0, 0, 1), 1),
((0, 1, 0), 1),
((0, 1, 1), 1),
...
((2, 1, 3), 1),
((2, 2, 1), 1)]
"""
u, v, s = self.__P1.normalize_with_scalar(x[1], x[2])
return (x[0], u, v), self.__character(s)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList', ManinSymbolList)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList_group', ManinSymbolList_group)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList_gamma0', ManinSymbolList_gamma0)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList_gamma1', ManinSymbolList_gamma1)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList_gamma_h', ManinSymbolList_gamma_h)
register_unpickle_override('sage.modular.modsym.manin_symbols',
'ManinSymbolList_character',
ManinSymbolList_character)
INPUT:
- ``self`` -- an instance of the LLT hcospin basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a partition and returns the coefficient
in the expansion of the monomial basis
EXAMPLES::
sage: HCosp3 = SymmetricFunctions(FractionField(QQ['t'])).llt(3).hcospin()
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
sage: HCosp3.symmetric_function_ring().m()( HCosp3[2,1] )
(2*t+1)*m[1, 1, 1] + (t+1)*m[2, 1] + m[3]
"""
level = self.level()
f = lambda part2: QQt(ribbon_tableau.cospin_polynomial([level*i for i in part], part2, level))
return f
class Element(LLT_generic.Element):
pass
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_spin', LLT_spin.Element)
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_cospin', LLT_cospin.Element)
"""
for p in descents_composition_list(self.shape):
yield self.from_permutation(p)
class Ribbon_class(RibbonShapedTableau):
"""
This exists solely for unpickling ``Ribbon_class`` objects.
"""
def __setstate__(self, state):
r"""
Unpickle old ``Ribbon_class`` objects.
EXAMPLES::
sage: loads(b'x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1+\xcaLJ\xca\xcf\xe3\n\x02S\xf1\xc99\x89\xc5\xc5\\\x85\x8c\x9a\x8d\x85L\xb5\x85\xcc\x1a\xa1\xac\xf1\x19\x89\xc5\x19\x85,~@VNfqI!kl!\x9bFl!\xbb\x06\xc4\x9c\xa2\xcc\xbc\xf4b\xbd\xcc\xbc\x92\xd4\xf4\xd4"\xae\xdc\xc4\xec\xd4x\x18\xa7\x90#\x94\xd1\xb05\xa8\x903\x03\xc80\x022\xb8Rc\x0b\xb95@<c \x8f\x07\xc40\x012xSSK\x93\xf4\x00l\x811\x17')
[[None, 1, 2], [3, 4]]
sage: loads(dumps( RibbonShapedTableau([[3,2,1], [1,1]]) )) # indirect doctest
[[None, 3, 2, 1], [1, 1]]
"""
self.__class__ = RibbonShapedTableau
self.__init__(RibbonShapedTableaux(), state['_list'])
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.ribbon', 'Ribbon_class',
Ribbon_class)
register_unpickle_override('sage.combinat.ribbon', 'StandardRibbons_shape',
StandardRibbonShapedTableaux)
sage: all = iter(K)
sage: next(all)
0
sage: next(all)
1
sage: next(all)
2
"""
yield self(0)
i = one = self(1)
while i:
yield i
i += one
def degree(self):
"""
Return the degree of ``self`` over its prime field.
This always returns 1.
EXAMPLES::
sage: FiniteField(3).degree()
1
"""
return Integer(1)
register_unpickle_override(
'sage.rings.finite_field_prime_modn', 'FiniteField_prime_modn',
FiniteField_prime_modn)
if implementation is None:
implementation = "matrix"
if implementation == "reflection":
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError("coxeter3 must be installed")
else:
return CoxeterGroup(cartan_type)
if implementation == "permutation":
if not cartan_type.is_finite():
raise ValueError("the type must be finite")
if cartan_type.is_crystallographic():
return WeylGroup(cartan_type, implementation="permutation")
return ReflectionGroup(cartan_type, index_set=index_set)
elif implementation == "matrix":
if cartan_type.is_crystallographic():
return WeylGroup(cartan_type)
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
elif implementation == "chevie":
return ReflectionGroup(cartan_type, index_set=index_set)
raise NotImplementedError("Coxeter group of type {} as {} group not implemented".format(cartan_type, implementation))
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.root_system.coxeter_group', 'CoxeterGroupAsPermutationGroup', ReflectionGroup)
def _multiply(self, left, right):
"""
Returns ``left`` * ``right`` by converting both to the base and then
converting back to ``self``.
INPUT:
- ``self`` -- a basis determined by an orthotriangular definition
- ``left``, ``right`` -- elements in ``self``
OUTPUT:
- the expansion of the product of ``left`` and ``right`` in the basis ``self``.
EXAMPLES::
sage: from sage.combinat.sf.sfa import zee
sage: from sage.combinat.sf.orthotriang import SymmetricFunctionAlgebra_orthotriang
sage: Sym = SymmetricFunctions(QQ)
sage: m = Sym.m()
sage: s = SymmetricFunctionAlgebra_orthotriang(Sym, m, zee, 's', 'Schur functions')
sage: s([1])*s([2,1]) #indirect doctest
s[2, 1, 1] + s[2, 2] + s[3, 1]
"""
return self( self._sf_base(left)*self._sf_base(right) )
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.orthotriang', 'SymmetricFunctionAlgebraElement_orthotriang', SymmetricFunctionAlgebra_orthotriang.Element)
sage: from sage.combinat.words.alphabet import OrderedAlphabet
sage: O = OrderedAlphabet()
doctest:...: DeprecationWarning: OrderedAlphabet is deprecated; use Alphabet instead.
See http://trac.sagemath.org/8920 for details.
sage: O._alphabet = ['a', 'b']
sage: O._elements
('a', 'b')
"""
if name == '_elements':
if not hasattr(self, '_alphabet'):
raise AttributeError("no attribute '_elements'")
self._elements = tuple(self._alphabet)
from sage.structure.parent import Parent
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
Parent.__init__(self, category=FiniteEnumeratedSets(), facade=True)
return self._elements
raise AttributeError("no attribute %s" % name)
register_unpickle_override(
'sage.combinat.words.alphabet',
'OrderedAlphabet_NaturalNumbers',
NonNegativeIntegers,
call_name=('sage.sets.non_negative_integers', 'NonNegativeIntegers'))
register_unpickle_override(
'sage.combinat.words.alphabet',
'OrderedAlphabet_PositiveIntegers',
PositiveIntegers,
call_name=('sage.sets.positive_integers', 'PositiveIntegers'))
from __future__ import absolute_import
from .base import IntegerListsBackend, Envelope
from .lists import IntegerLists
from .invlex import IntegerListsLex
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.integer_list', 'IntegerListsLex', IntegerListsLex)
OUTPUT:
- an element of the base ring
EXAMPLES::
sage: p = SymmetricFunctions(QQ).p()
sage: p([3,3]).eval_at_permutation_roots([6])
0
sage: p([3,3]).eval_at_permutation_roots([3])
9
sage: p([3,3]).eval_at_permutation_roots([1])
1
sage: p([3,3]).eval_at_permutation_roots([3,3])
36
sage: p([3,3]).eval_at_permutation_roots([1,1,1,1,1])
25
sage: (p[1]+p[2]+p[3]).eval_at_permutation_roots([3,2])
5
"""
p = self.parent()
R = self.base_ring()
on_basis = lambda lam: R.prod(
p.eval_at_permutation_roots_on_generators(k, rho) for k in lam)
return p._apply_module_morphism(self, on_basis, R)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.powersum', 'SymmetricFunctionAlgebraElement_power', SymmetricFunctionAlgebra_power.Element)
def _derivative_(self, x, diff_param=None):
"""
Derivative of inverse erf function.
EXAMPLES::
sage: erfinv(x).diff(x)
1/2*sqrt(pi)*e^(erfinv(x)^2)
"""
return sqrt(pi)*exp(erfinv(x)**2)/2
erfinv = Function_erfinv()
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.functions.other', 'Function_erf', Function_erf)
############################
# Fresnel integrals #
############################
class Function_Fresnel_sin(BuiltinFunction):
def __init__(self):
r"""
The sine Fresnel integral.
It is defined by the integral
.. MATH ::
\operatorname{S}(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right)\, dt
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 __iter__(self):
"""
TESTS::
sage: SignedCompositions(0).list() #indirect doctest
[[]]
sage: SignedCompositions(1).list() #indirect doctest
[[1], [-1]]
sage: SignedCompositions(2).list() #indirect doctest
[[1, 1], [1, -1], [-1, 1], [-1, -1], [2], [-2]]
"""
for comp in Compositions_n.__iter__(self):
l = len(comp)
for sign in itertools.product([1,-1], repeat=l):
yield [ sign[i]*comp[i] for i in range(l)]
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.composition_signed', 'SignedCompositions_n', SignedCompositions)
def unpickle_PolynomialRing(base_ring, arg1=None, arg2=None, sparse=False):
"""
Custom unpickling function for polynomial rings.
This has the same positional arguments as the old
``PolynomialRing`` constructor before :trac:`23338`.
"""
args = [arg for arg in (arg1, arg2) if arg is not None]
return PolynomialRing(base_ring, *args, sparse=sparse)
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.rings.polynomial.polynomial_ring_constructor',
'PolynomialRing', unpickle_PolynomialRing)
def _get_from_cache(key):
key = tuple(key)
return _cache.get(key)
def _save_in_cache(key, R):
key = tuple(key)
_cache[key] = R
def _single_variate(base_ring,
name,
sparse=None,
b = b - 1
for i in range(k):
comb[i] = (n - 1) - comb[i]
return tuple(comb)
##########################################################
# Deprecations
class ChooseNK(Combinations_setk):
def __setstate__(self, state):
r"""
For unpickling old ``ChooseNK`` objects.
TESTS::
sage: loads(b"x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1K\xce\xc8\xcf"
....: b"/N\x8d\xcf\xcb\xe6r\x06\xb3\xfc\xbc\xb9\n\x195\x1b\x0b"
....: b"\x99j\x0b\x995B\x99\xe2\xf3\nY :\x8a2\xf3\xd2\x8b\xf52"
....: b"\xf3JR\xd3S\x8b\xb8r\x13\xb3S\xe3a\x9cB\xd6PF\xd3\xd6\xa0"
....: b"B6\xa0\xfa\xecB\xf6\x0c \xd7\x08\xc8\xe5(M\xd2\x03\x00{"
....: b"\x82$\xd8")
Combinations of [0, 1, 2, 3, 4] of length 2
"""
self.__class__ = Combinations_setk
Combinations_setk.__init__(self, list(range(state['_n'])), state['_k'])
from sage.misc.persist import register_unpickle_override
register_unpickle_override("sage.combinat.choose_nk", "ChooseNK", ChooseNK)
"""
for p in descents_composition_list(self.shape):
yield self.from_permutation(p)
class Ribbon_class(RibbonShapedTableau):
"""
This exists solely for unpickling ``Ribbon_class`` objects.
"""
def __setstate__(self, state):
r"""
Unpickle old ``Ribbon_class`` objects.
EXAMPLES::
sage: loads(b'x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1+\xcaLJ\xca\xcf\xe3\n\x02S\xf1\xc99\x89\xc5\xc5\\\x85\x8c\x9a\x8d\x85L\xb5\x85\xcc\x1a\xa1\xac\xf1\x19\x89\xc5\x19\x85,~@VNfqI!kl!\x9bFl!\xbb\x06\xc4\x9c\xa2\xcc\xbc\xf4b\xbd\xcc\xbc\x92\xd4\xf4\xd4"\xae\xdc\xc4\xec\xd4x\x18\xa7\x90#\x94\xd1\xb05\xa8\x903\x03\xc80\x022\xb8Rc\x0b\xb95@<c \x8f\x07\xc40\x012xSSK\x93\xf4\x00l\x811\x17')
[[None, 1, 2], [3, 4]]
sage: loads(dumps( RibbonShapedTableau([[3,2,1], [1,1]]) )) # indirect doctest
[[None, 3, 2, 1], [1, 1]]
"""
self.__class__ = RibbonShapedTableau
self.__init__(RibbonShapedTableaux(), state['_list'])
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.ribbon', 'Ribbon_class', Ribbon_class)
register_unpickle_override('sage.combinat.ribbon', 'StandardRibbons_shape', StandardRibbonShapedTableaux)
# Deprecations from trac:18555. July 2016
from sage.misc.superseded import deprecated_function_alias
RibbonShapedTableaux.global_options = deprecated_function_alias(18555, RibbonShapedTableaux.options)
StandardRibbonShapedTableaux.global_options = deprecated_function_alias(18555, StandardRibbonShapedTableaux.options)
import sage.libs.flint.flint
sage.libs.flint.flint.free_flint_stack()
# Free globally allocated mpir integers.
import sage.rings.integer
sage.rings.integer.free_integer_pool()
import sage.algebras.quatalg.quaternion_algebra_element
sage.algebras.quatalg.quaternion_algebra_element._clear_globals()
from sage.libs.all import symmetrica
symmetrica.end()
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.categories.category', 'Sets', Sets)
register_unpickle_override('sage.categories.category_types', 'HeckeModules', HeckeModules)
register_unpickle_override('sage.categories.category_types', 'Objects', Objects)
register_unpickle_override('sage.categories.category_types', 'Rings', Rings)
register_unpickle_override('sage.categories.category_types', 'Fields', Fields)
register_unpickle_override('sage.categories.category_types', 'VectorSpaces', VectorSpaces)
register_unpickle_override('sage.categories.category_types', 'Schemes_over_base', sage.categories.schemes.Schemes_over_base)
register_unpickle_override('sage.categories.category_types', 'ModularAbelianVarieties', ModularAbelianVarieties)
register_unpickle_override('sage.libs.pari.gen_py', 'pari', pari)
# Cache the contents of star imports.
sage.misc.lazy_import.save_cache_file()
### Debugging for Singular, see trac #10903
# from sage.libs.singular.ring import poison_currRing
##########################################################
# Deprecations
class SplitNK(OrderedSetPartitions_scomp):
def __setstate__(self, state):
r"""
For unpickling old ``SplitNK`` objects.
TESTS::
sage: loads(b"x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1+.\xc8\xc9,"
....: b"\x89\xcf\xcb\xe6\n\x061\xfc\xbcA\xccBF\xcd\xc6B\xa6\xda"
....: b"Bf\x8dP\xa6\xf8\xbcB\x16\x88\x96\xa2\xcc\xbc\xf4b\xbd\xcc"
....: b"\xbc\x92\xd4\xf4\xd4\"\xae\xdc\xc4\xec\xd4x\x18\xa7\x905"
....: b"\x94\xd1\xb45\xa8\x90\r\xa8>\xbb\x90=\x03\xc85\x02r9J\x93"
....: b"\xf4\x00\xb4\xc6%f")
Ordered set partitions of {0, 1, 2, 3, 4} into parts of size [2, 3]
"""
self.__class__ = OrderedSetPartitions_scomp
n = state['_n']
k = state['_k']
OrderedSetPartitions_scomp.__init__(self, range(state['_n']),
(k, n - k))
from sage.misc.persist import register_unpickle_override
register_unpickle_override("sage.combinat.split_nk", "SplitNK_nk", SplitNK)
EXAMPLES::
sage: h = SymmetricFunctions(QQ).h()
sage: h([3]).expand(2)
x0^3 + x0^2*x1 + x0*x1^2 + x1^3
sage: h([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: h([2,1]).expand(3)
x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3
sage: h([3]).expand(2,alphabet='y')
y0^3 + y0^2*y1 + y0*y1^2 + y1^3
sage: h([3]).expand(2,alphabet='x,y')
x^3 + x^2*y + x*y^2 + y^3
sage: h([3]).expand(3,alphabet='x,y,z')
x^3 + x^2*y + x*y^2 + y^3 + x^2*z + x*y*z + y^2*z + x*z^2 + y*z^2 + z^3
sage: (h([]) + 2*h([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
"""
if n == 0: # Symmetrica crashes otherwise...
return self.counit()
condition = lambda part: False
return self._expand(condition, n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.homogeneous', 'SymmetricFunctionAlgebraElement_homogeneous', SymmetricFunctionAlgebra_homogeneous.Element)
OUTPUT:
A monomial expansion of ``self`` in the `n` variables
labelled by ``alphabet``.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(zee)
sage: a = h([2,1])+h([3])
sage: a.expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.dual().expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.expand(2,alphabet='y')
2*y0^3 + 3*y0^2*y1 + 3*y0*y1^2 + 2*y1^3
sage: a.expand(2,alphabet='x,y')
2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
"""
return self._dual.expand(n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.dual', 'SymmetricFunctionAlgebraElement_dual', SymmetricFunctionAlgebra_dual.Element)
OUTPUT:
A monomial expansion of ``self`` in the `n` variables
labelled by ``alphabet``.
EXAMPLES::
sage: m = SymmetricFunctions(QQ).monomial()
sage: zee = sage.combinat.sf.sfa.zee
sage: h = m.dual_basis(zee)
sage: a = h([2,1])+h([3])
sage: a.expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.dual().expand(2)
2*x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + 2*x1^3
sage: a.expand(2,alphabet='y')
2*y0^3 + 3*y0^2*y1 + 3*y0*y1^2 + 2*y1^3
sage: a.expand(2,alphabet='x,y')
2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
"""
return self._dual.expand(n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.dual', 'SymmetricFunctionAlgebraElement_dual', SymmetricFunctionAlgebra_dual.Element)
import sage.libs.flint.flint
sage.libs.flint.flint.free_flint_stack()
# Free globally allocated mpir integers.
import sage.rings.integer
sage.rings.integer.free_integer_pool()
import sage.algebras.quatalg.quaternion_algebra_element
sage.algebras.quatalg.quaternion_algebra_element._clear_globals()
from sage.libs.all import symmetrica
symmetrica.end()
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.categories.category', 'Sets', Sets)
register_unpickle_override('sage.categories.category_types', 'HeckeModules', HeckeModules)
register_unpickle_override('sage.categories.category_types', 'Objects', Objects)
register_unpickle_override('sage.categories.category_types', 'Rings', Rings)
register_unpickle_override('sage.categories.category_types', 'Fields', Fields)
register_unpickle_override('sage.categories.category_types', 'VectorSpaces', VectorSpaces)
register_unpickle_override('sage.categories.category_types', 'Schemes_over_base', sage.categories.schemes.Schemes_over_base)
register_unpickle_override('sage.categories.category_types', 'ModularAbelianVarieties', ModularAbelianVarieties)
register_unpickle_override('sage.libs.pari.gen_py', 'pari', pari)
# Cache the contents of star imports.
sage.misc.lazy_import.save_cache_file()
### Debugging for Singular, see trac #10903
# from sage.libs.singular.ring import poison_currRing
from __future__ import absolute_import
import sage.crypto.sbox
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.crypto.mq.sbox', 'SBox', sage.crypto.sbox.SBox)
from sage.misc.lazy_import import lazy_import
lazy_import('sage.crypto.classical', ['AffineCryptosystem',
'HillCryptosystem',
'SubstitutionCryptosystem',
'ShiftCryptosystem',
'TranspositionCryptosystem',
'VigenereCryptosystem',
])
lazy_import('sage.crypto.stream', ['LFSRCryptosystem',
'ShrinkingGeneratorCryptosystem',
])
lazy_import('sage.crypto.lfsr', ['lfsr_sequence',
'lfsr_autocorrelation',
'lfsr_connection_polynomial',
])
sage: h([3]).expand(2)
x0^3 + x0^2*x1 + x0*x1^2 + x1^3
sage: h([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: h([2,1]).expand(3)
x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3
sage: h([3]).expand(2,alphabet='y')
y0^3 + y0^2*y1 + y0*y1^2 + y1^3
sage: h([3]).expand(2,alphabet='x,y')
x^3 + x^2*y + x*y^2 + y^3
sage: h([3]).expand(3,alphabet='x,y,z')
x^3 + x^2*y + x*y^2 + y^3 + x^2*z + x*y*z + y^2*z + x*z^2 + y*z^2 + z^3
sage: (h([]) + 2*h([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
"""
if n == 0: # Symmetrica crashes otherwise...
return self.counit()
condition = lambda part: False
return self._expand(condition, n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.homogeneous',
'SymmetricFunctionAlgebraElement_homogeneous',
SymmetricFunctionAlgebra_homogeneous.Element)
labelled by ``alphabet``.
EXAMPLES::
sage: e = SymmetricFunctions(QQ).e()
sage: e([2,1]).expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 3*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: e([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: e([3]).expand(2)
0
sage: e([2]).expand(3)
x0*x1 + x0*x2 + x1*x2
sage: e([3]).expand(4,alphabet='x,y,z,t')
x*y*z + x*y*t + x*z*t + y*z*t
sage: e([3]).expand(4,alphabet='y')
y0*y1*y2 + y0*y1*y3 + y0*y2*y3 + y1*y2*y3
sage: e([]).expand(2)
1
sage: e([]).expand(0)
1
sage: (3*e([])).expand(0)
3
"""
condition = lambda part: max(part) > n
return self._expand(condition, n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.elementary', 'SymmetricFunctionAlgebraElement_elementary', SymmetricFunctionAlgebra_elementary.Element)
D[j] = e
j += 1
else:
D[k - 1] = n0 + r + k - 1 - B
return tuple(D)
##########################################################
# Deprecations
class ChooseNK(Combinations_setk):
def __setstate__(self, state):
r"""
For unpickling old ``ChooseNK`` objects.
TESTS::
sage: loads(b"x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1K\xce\xc8\xcf"
....: b"/N\x8d\xcf\xcb\xe6r\x06\xb3\xfc\xbc\xb9\n\x195\x1b\x0b"
....: b"\x99j\x0b\x995B\x99\xe2\xf3\nY :\x8a2\xf3\xd2\x8b\xf52"
....: b"\xf3JR\xd3S\x8b\xb8r\x13\xb3S\xe3a\x9cB\xd6PF\xd3\xd6\xa0"
....: b"B6\xa0\xfa\xecB\xf6\x0c \xd7\x08\xc8\xe5(M\xd2\x03\x00{"
....: b"\x82$\xd8")
Combinations of [0, 1, 2, 3, 4] of length 2
"""
self.__class__ = Combinations_setk
Combinations_setk.__init__(self, list(range(state['_n'])), state['_k'])
register_unpickle_override("sage.combinat.choose_nk", "ChooseNK", ChooseNK)
EXAMPLES::
sage: s = SymmetricFunctions(QQ).s()
sage: a = s([2,1])
sage: a.expand(2)
x0^2*x1 + x0*x1^2
sage: a.expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: a.expand(4)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + x0^2*x3 + 2*x0*x1*x3 + x1^2*x3 + 2*x0*x2*x3 + 2*x1*x2*x3 + x2^2*x3 + x0*x3^2 + x1*x3^2 + x2*x3^2
sage: a.expand(2, alphabet='y')
y0^2*y1 + y0*y1^2
sage: a.expand(2, alphabet=['a','b'])
a^2*b + a*b^2
sage: s([1,1,1,1]).expand(3)
0
sage: (s([]) + 2*s([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: s([1]).expand(0)
0
sage: (3*s([])).expand(0)
3
"""
condition = lambda part: len(part) > n
return self._expand(condition, n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.schur', 'SymmetricFunctionAlgebraElement_schur', SymmetricFunctionAlgebra_schur.Element)
return
if len(l) == 1:
if n % l[0] == 0:
yield [n // l[0]]
return
k = 0
cur = [n // l[k] + one]
rem = n - cur[-1] * l[k] # Amount remaining
while cur:
cur[-1] -= one
rem += l[k]
if rem == zero:
yield cur + [zero] * (len(l) - len(cur))
elif cur[-1] < zero or rem < zero:
rem += cur.pop() * l[k]
k -= 1
elif len(l) == len(cur) + 1:
if rem % l[-1] == zero:
yield cur + [rem // l[-1]]
else:
k += 1
cur.append(rem // l[k] + one)
rem -= cur[-1] * l[k]
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.integer_vector_weighted',
'WeightedIntegerVectors_nweight',
WeightedIntegerVectors)
sage: a = s([2,1])
sage: a.expand(2)
x0^2*x1 + x0*x1^2
sage: a.expand(3)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2
sage: a.expand(4)
x0^2*x1 + x0*x1^2 + x0^2*x2 + 2*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 + x0^2*x3 + 2*x0*x1*x3 + x1^2*x3 + 2*x0*x2*x3 + 2*x1*x2*x3 + x2^2*x3 + x0*x3^2 + x1*x3^2 + x2*x3^2
sage: a.expand(2, alphabet='y')
y0^2*y1 + y0*y1^2
sage: a.expand(2, alphabet=['a','b'])
a^2*b + a*b^2
sage: s([1,1,1,1]).expand(3)
0
sage: (s([]) + 2*s([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: s([1]).expand(0)
0
sage: (3*s([])).expand(0)
3
"""
condition = lambda part: len(part) > n
return self._expand(condition, n, alphabet)
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.schur',
'SymmetricFunctionAlgebraElement_schur',
SymmetricFunctionAlgebra_schur.Element)
sage: from sage.combinat.words.alphabet import OrderedAlphabet
sage: O = OrderedAlphabet()
doctest:...: DeprecationWarning: OrderedAlphabet is deprecated; use Alphabet instead.
See http://trac.sagemath.org/8920 for details.
sage: O._alphabet = ['a', 'b']
sage: O._elements
('a', 'b')
"""
if name == '_elements':
if not hasattr(self, '_alphabet'):
raise AttributeError("no attribute '_elements'")
self._elements = tuple(self._alphabet)
from sage.structure.parent import Parent
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
Parent.__init__(self, category=FiniteEnumeratedSets(), facade=True)
return self._elements
raise AttributeError("no attribute %s" % name)
register_unpickle_override('sage.combinat.words.alphabet',
'OrderedAlphabet_NaturalNumbers',
NonNegativeIntegers,
call_name=('sage.sets.non_negative_integers',
'NonNegativeIntegers'))
register_unpickle_override('sage.combinat.words.alphabet',
'OrderedAlphabet_PositiveIntegers',
PositiveIntegers,
call_name=('sage.sets.positive_integers',
'PositiveIntegers'))
if extended is not self:
action = PrecomposedAction(
action, None, extended._internal_coerce_map_from(self))
return action
def _an_element_(self, check=False, reduce=False):
"""
EXAMPLES::
sage: FormalSums(ZZ).an_element() # indirect test
1
sage: FormalSums(QQ).an_element()
1/2*1
sage: QQ.an_element()
1/2
"""
return self.element_class([(self.base_ring().an_element(), 1)],
check=check,
reduce=reduce,
parent=self)
formal_sums = FormalSums()
# Formal sums now derives from UniqueRepresentation, which makes the
# factory function unnecessary. This is why the name was changed from
# class FormalSums_generic to class FormalSums.
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.structure.formal_sum', 'FormalSums_generic',
FormalSums)
sage: Hom(GF(3^3, 'a'), GF(3^6, 'b')).an_element()
Ring morphism:
From: Finite Field in a of size 3^3
To: Finite Field in b of size 3^6
Defn: a |--> 2*b^5 + 2*b^4
sage: Hom(GF(3^3, 'a'), GF(3^2, 'c')).an_element()
Traceback (most recent call last):
...
EmptySetError: no homomorphisms from Finite Field in a of size 3^3 to Finite Field in c of size 3^2
.. TODO::
Use a more sophisticated algorithm; see also :trac:`8751`.
"""
K = self.domain()
L = self.codomain()
if K.degree() == 1:
return L.coerce_map_from(K)
elif not K.degree().divides(L.degree()):
from sage.categories.sets_cat import EmptySetError
raise EmptySetError('no homomorphisms from %s to %s' % (K, L))
return K.hom([K.modulus().any_root(L)])
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.rings.finite_field_morphism',
'FiniteFieldHomset', FiniteFieldHomset)
def _derivative_(self, x, diff_param=None):
"""
Derivative of inverse erf function.
EXAMPLES::
sage: erfinv(x).diff(x)
1/2*sqrt(pi)*e^(erfinv(x)^2)
"""
return sqrt(pi) * exp(erfinv(x)**2) / 2
erfinv = Function_erfinv()
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.functions.other', 'Function_erf',
Function_erf)
############################
# Fresnel integrals #
############################
class Function_Fresnel_sin(BuiltinFunction):
def __init__(self):
r"""
The sine Fresnel integral.
It is defined by the integral
.. MATH ::
\operatorname{S}(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right)\, dt
Return ``left`` * ``right`` by converting both to the base and then
converting back to ``self``.
INPUT:
- ``self`` -- a basis determined by an orthotriangular definition
- ``left``, ``right`` -- elements in ``self``
OUTPUT:
- the expansion of the product of ``left`` and ``right`` in the basis ``self``.
EXAMPLES::
sage: from sage.combinat.sf.sfa import zee
sage: from sage.combinat.sf.orthotriang import SymmetricFunctionAlgebra_orthotriang
sage: Sym = SymmetricFunctions(QQ)
sage: m = Sym.m()
sage: s = SymmetricFunctionAlgebra_orthotriang(Sym, m, zee, 's', 'Schur functions')
sage: s([1])*s([2,1]) #indirect doctest
s[2, 1, 1] + s[2, 2] + s[3, 1]
"""
return self(self._sf_base(left) * self._sf_base(right))
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.orthotriang',
'SymmetricFunctionAlgebraElement_orthotriang',
SymmetricFunctionAlgebra_orthotriang.Element)
one = ZZ.one()
k = int(k)
pos = 0 # Current position
rem = zero # Amount remaining
cur = [n] + [zero] * (k - 1) # Current list
yield list(cur)
while pos >= 0:
if not cur[pos]:
pos -= 1
continue
cur[pos] -= one
rem += one
if not rem:
yield list(cur)
elif pos == k - 2:
cur[pos + 1] = rem
yield list(cur)
cur[pos + 1] = zero
else:
pos += 1
cur[pos] = rem # Guaranteed to be at least 1
rem = zero
yield list(cur)
# October 2012: fixing outdated pickles which use classes being deprecated
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.integer_vector', 'IntegerVectors_nconstraints', IntegerVectorsConstraints)
register_unpickle_override('sage.combinat.integer_vector', 'IntegerVectors_nkconstraints', IntegerVectorsConstraints)
import sage.crypto.sbox
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.crypto.mq.sbox', 'SBox',
sage.crypto.sbox.SBox)
from sage.misc.lazy_import import lazy_import
lazy_import('sage.crypto.classical', [
'AffineCryptosystem',
'HillCryptosystem',
'SubstitutionCryptosystem',
'ShiftCryptosystem',
'TranspositionCryptosystem',
'VigenereCryptosystem',
])
lazy_import('sage.crypto.stream', [
'LFSRCryptosystem',
'ShrinkingGeneratorCryptosystem',
])
lazy_import('sage.crypto.lfsr', [
'lfsr_sequence',
'lfsr_autocorrelation',
'lfsr_connection_polynomial',
])
EXAMPLES::
sage: R = Integers(12345678900)
sage: R.degree()
1
"""
return integer.Integer(1)
Zmod = IntegerModRing
Integers = IntegerModRing
# Register unpickling methods for backward compatibility.
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.rings.integer_mod_ring',
'IntegerModRing_generic', IntegerModRing_generic)
def crt(v):
"""
INPUT:
- ``v`` -- (list) a lift of elements of ``rings.IntegerMod(n)``, for
various coprime moduli ``n``
EXAMPLES::
sage: from sage.rings.finite_rings.integer_mod_ring import crt
sage: crt([mod(3, 8),mod(1,19),mod(7, 15)])
1027
"""
1 2
sage: print(CartanType(['C',3]).ascii_art())
O---O=<=O
1 2 3
sage: print(CartanType(['C',5]).ascii_art(label = lambda x: x+2))
O---O---O---O=<=O
3 4 5 6 7
"""
return self.dual().ascii_art(label=label,
node=node).replace("=>=", "=<=")
def _default_folded_cartan_type(self):
"""
Return the default folded Cartan type.
EXAMPLES::
sage: CartanType(['C', 3])._default_folded_cartan_type()
['C', 3] as a folding of ['A', 5]
"""
from sage.combinat.root_system.type_folded import CartanTypeFolded
n = self.n
return CartanTypeFolded(self, ['A', 2 * n - 1],
[[i, 2 * n - i] for i in range(1, n)] + [[n]])
# For unpickling backward compatibility (Sage <= 4.1)
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.root_system.type_C', 'ambient_space',
AmbientSpace)
EXAMPLES::
sage: print(CartanType(['E',6]).ascii_art(label = lambda x: x+2))
O 4
|
|
O---O---O---O---O
3 5 6 7 8
sage: print(CartanType(['E',7]).ascii_art(label = lambda x: x+2))
O 4
|
|
O---O---O---O---O---O
3 5 6 7 8 9
sage: print(CartanType(['E',8]).ascii_art(label = lambda x: x+1))
O 3
|
|
O---O---O---O---O---O---O
2 4 5 6 7 8 9
"""
if node is None:
node = self._ascii_art_node
labels = [label(_) for _ in [1,3,4,5,6] + list(range(7, self.n+1))] # We exclude 2 because of the special case
ret = " {} {}\n |\n |\n".format(node(label(2)), label(2))
return ret + '---'.join(node(i) for i in labels) + '\n' + "".join("{!s:4}".format(i) for i in labels)
# For unpickling backward compatibility (Sage <= 4.1)
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.root_system.type_E', 'ambient_space', AmbientSpace)
"""
if name == "_Sequence_generic__cr" and hasattr(self, "_Sequence__cr"):
self.__cr = self._Sequence__cr
return self.__cr
elif name == "_Sequence_generic__cr_str" and hasattr(
self, "_Sequence__cr_str"):
self.__cr_str = self._Sequence__cr_str
return self.__cr_str
elif name == "_Sequence_generic__immutable" and hasattr(
self, "_Sequence__immutable"):
self.__immutable = self._Sequence__immutable
return self.__immutable
elif name == "_Sequence_generic__universe" and hasattr(
self, "_Sequence__universe"):
self.__universe = self._Sequence__universe
return self.__universe
elif name == "_Sequence_generic__hash" and hasattr(
self, "_Sequence__hash"):
self.__hash = self._Sequence__hash
return self.__hash
else:
raise AttributeError(
"'Sequence_generic' object has no attribute '%s'" % name)
seq = Sequence
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.structure.sequence', 'Sequence',
Sequence_generic)
from .base import IntegerListsBackend, Envelope
from .lists import IntegerLists
from .invlex import IntegerListsLex
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.integer_list', 'IntegerListsLex', IntegerListsLex)
from .chain_complex import ChainComplex
from .chain_complex_morphism import ChainComplexMorphism
from .simplicial_complex import SimplicialComplex, Simplex
from .simplicial_complex_morphism import SimplicialComplexMorphism
from .delta_complex import DeltaComplex, delta_complexes
from .cubical_complex import CubicalComplex, cubical_complexes
from sage.misc.lazy_import import lazy_import
lazy_import('sage.homology.koszul_complex', 'KoszulComplex')
lazy_import('sage.homology', 'simplicial_complexes_catalog',
'simplicial_complexes')
lazy_import('sage.homology', 'simplicial_set_catalog', 'simplicial_sets')
# For taking care of old pickles
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.homology.examples', 'SimplicialSurface',
SimplicialComplex)
TESTS::
sage: P2.<x,y,z> = AffineSpace(3, GF(3))
sage: point_homset = P2._point_homset(Spec(GF(3)), P2)
sage: P2._point(point_homset, [1, 2, 3])
(1, 2, 0)
"""
return SchemeMorphism_point_affine_finite_field(*args, **kwds)
def _morphism(self, *args, **kwds):
"""
Construct a morphism.
For internal use only. See :mod:`morphism` for details.
TESTS::
sage: P2.<x,y,z> = AffineSpace(3, GF(3))
sage: P2._morphism(P2.Hom(P2), [x, y, z])
Scheme endomorphism of Affine Space of dimension 3 over Finite Field of size 3
Defn: Defined on coordinates by sending (x, y, z) to
(x, y, z)
"""
return SchemeMorphism_polynomial_affine_space_finite_field(*args, **kwds)
#fix the pickles from moving affine_space.py
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.schemes.generic.affine_space',
'AffineSpace_generic',
AffineSpace_generic)
sage: from sage.modular.modsym.g1list import _G1list_old_pickle
sage: L = _G1list_old_pickle()
sage: type(L) == G1list
True
"""
self.__class__ = G1list
def __setstate__(self, state):
"""
For unpickling new pickles.
TESTS::
sage: from sage.modular.modsym.g1list import G1list
sage: L = G1list(6)
sage: Lp = loads(dumps(L))
sage: L == Lp
True
sage: type(Lp) == G1list
True
"""
# We don't really want this class, but we want to handle new
# pickles without creating a new class
self.__class__ = G1list
self.__dict__ = state # Default pickling is ``state = self.__dict__``
register_unpickle_override('sage.modular.modsym.g1list', 'G1list',
_G1list_old_pickle)
INPUT:
- ``self`` -- an instance of the LLT hcospin basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a partition and returns the coefficient
in the expansion of the monomial basis
EXAMPLES::
sage: HCosp3 = SymmetricFunctions(FractionField(QQ['t'])).llt(3).hcospin()
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
sage: HCosp3.symmetric_function_ring().m()( HCosp3[2,1] )
(2*t+1)*m[1, 1, 1] + (t+1)*m[2, 1] + m[3]
"""
level = self.level()
f = lambda part2: QQt(ribbon_tableau.cospin_polynomial([level*i for i in part], part2, level))
return f
class Element(LLT_generic.Element):
pass
# Backward compatibility for unpickling
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_spin', LLT_spin.Element)
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_cospin', LLT_cospin.Element)
to the method
EXAMPLES::
sage: f = attrcall('core', 3); f
*.core(3)
sage: [f(p) for p in Partitions(5)]
[[2], [1, 1], [1, 1], [3, 1, 1], [2], [2], [1, 1]]
"""
return AttrCallObject(name, args, kwds)
def call_method(obj, name, *args, **kwds):
"""
Call the method ``name`` on ``obj``.
This has to exist somewhere in Python!!!
.. SEEALSO:: :func:`operator.methodcaller` :func:`attrcal`
EXAMPLES::
sage: from sage.misc.call import call_method
sage: call_method(1, "__add__", 2)
3
"""
return getattr(obj, name)(*args, **kwds)
from sage.misc.persist import register_unpickle_override
register_unpickle_override("sage.misc.misc", "call_method", call_method)
"""
return richcmp([sorted(s) for s in left], [sorted(s) for s in right], op)
##########################################################
# Deprecations
class SplitNK(OrderedSetPartitions_scomp):
def __setstate__(self, state):
r"""
For unpickling old ``SplitNK`` objects.
TESTS::
sage: loads(b"x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\xd1+.\xc8\xc9,"
....: b"\x89\xcf\xcb\xe6\n\x061\xfc\xbcA\xccBF\xcd\xc6B\xa6\xda"
....: b"Bf\x8dP\xa6\xf8\xbcB\x16\x88\x96\xa2\xcc\xbc\xf4b\xbd\xcc"
....: b"\xbc\x92\xd4\xf4\xd4\"\xae\xdc\xc4\xec\xd4x\x18\xa7\x905"
....: b"\x94\xd1\xb45\xa8\x90\r\xa8>\xbb\x90=\x03\xc85\x02r9J\x93"
....: b"\xf4\x00\xb4\xc6%f")
Ordered set partitions of {0, 1, 2, 3, 4} into parts of size [2, 3]
"""
self.__class__ = OrderedSetPartitions_scomp
n = state['_n']
k = state['_k']
OrderedSetPartitions_scomp.__init__(self, range(state['_n']), (k,n-k))
from sage.misc.persist import register_unpickle_override
register_unpickle_override("sage.combinat.split_nk", "SplitNK_nk", SplitNK)