QSymExplorationsAtICERM
system:sage


{{{id=1|
QSym = QuasiSymmetricFunctions(QQ)
///
}}}

{{{id=2|
QSym
///
Quasisymmetric functions over the Rational Field
}}}

{{{id=3|
QSym.inject_shorthands()
///
Injecting M as shorthand for Quasisymmetric functions over the Rational Field in the Monomial basis
Injecting F as shorthand for Quasisymmetric functions over the Rational Field in the Fundamental basis
Injecting dI as shorthand for Quasisymmetric functions over the Rational Field in the dualImmaculate basis
}}}

{{{id=4|
x = M[2,1]
///
}}}

{{{id=6|
dict(x)
///
{[2, 1]: 1}
}}}

{{{id=7|
x.__dict__
///
{'_monomial_coefficients': {[2, 1]: 1}}
}}}

{{{id=8|
y = F(x)
y
///
-F[1, 1, 1] + F[2, 1]
}}}

{{{id=9|
y.__dict__
///
{'_monomial_coefficients': {[1, 1, 1]: -1, [2, 1]: 1}}
}}}

{{{id=10|
F(y)
///
-F[1, 1, 1] + F[2, 1]
}}}

{{{id=11|
y * 2
///
-2*F[1, 1, 1] + 2*F[2, 1]
}}}

{{{id=12|
y.parent??
///
<html><!--notruncate-->

<div class="docstring">
    
  <p><strong>File:</strong> /home/saliola/Applications/sage-5.6/devel/sage/sage/structure/element.pyx</p>
<p><strong>Source Code</strong> (starting at line 550):</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="k">def</span> <span class="nf">parent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
    <span class="sd">&quot;&quot;&quot;</span>
<span class="sd">    Returns parent of this element; or, if the optional argument x is</span>
<span class="sd">    supplied, the result of coercing x into the parent of this element.</span>
<span class="sd">    &quot;&quot;&quot;</span>
    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parent</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>


</div>
</html>
}}}

{{{id=13|
SymmetricFunctions(QQ).inject_shorthands()
///
/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/sf.py:1075: RuntimeWarning: redefining global value `e`
  inject_variable(shorthand, getattr(self, shorthand)())
}}}

{{{id=14|
m[2,1]
///
m[2, 1]
}}}

{{{id=15|
M(m[2,1])
///
M[1, 2] + M[2, 1]
}}}

{{{id=16|
M(s[2,1])
///
2*M[1, 1, 1] + M[1, 2] + M[2, 1]
}}}

{{{id=17|
m(M[1,2] + M[2,1])
///
m[2, 1]
}}}

{{{id=18|
m(M[1,2])
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_41.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bShNWzEsMl0p"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmp4wUrv8/___code___.py", line 3, in <module>
    exec compile(u'm(M[_sage_const_1 ,_sage_const_2 ])' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "parent.pyx", line 804, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7305)
  File "morphism.pyx", line 278, in sage.categories.morphism.SetMorphism._call_ (sage/categories/morphism.c:4568)
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/categories/modules_with_basis.py", line 1748, in preimage
    raise ValueError, "%s is not in the image of %s"%(f, self)
ValueError: M[1, 2] is not in the image of Generic morphism:
  From: Symmetric Functions over Rational Field in the monomial basis
  To:   Quasisymmetric functions over the Rational Field in the Monomial basis
}}}

{{{id=19|
B = {0:[M.one()], 1:[M[1]]}
///
}}}

{{{id=20|
B
///
{0: [M[]], 1: [M[1]]}
}}}

{{{id=21|
for b in B.values():
    for bb in B.values():
        p = b * bb
        B[p.degree()] = p
///
}}}

{{{id=22|
B
///
{1: M[1], 2: 2*M[1, 1] + M[2]}
}}}

{{{id=24|

///
}}}

{{{id=23|

///
}}}

<p><span style="font-size: large;">Constructing 1 in QSym</span></p>

{{{id=25|
M[[]]
///
M[]
}}}

{{{id=26|
M.one()
///
M[]
}}}

{{{id=27|
M(1)
///
M[]
}}}

{{{id=29|

///
}}}

{{{id=28|

///
}}}

{{{id=30|
x = M[1,2]
///
}}}

{{{id=31|
x
///
M[1, 2]
}}}

{{{id=32|
show(x)
///
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}M_{[1, 2]}</script></html>
}}}

{{{id=33|
view(x)
///
<html><script type="math/tex">\newcommand{\Bold}[1]{\mathbf{#1}}M_{[1, 2]}</script></html>
}}}

{{{id=34|
view(x^3, viewer='pdf')
///
}}}

{{{id=35|
for c in Compositions(4):
    F.monomial(c)
///
F[1, 1, 1, 1]
F[1, 1, 2]
F[1, 2, 1]
F[1, 3]
F[2, 1, 1]
F[2, 2]
F[3, 1]
F[4]
}}}

<p><span style="font-size: large;">Convenient shorthand :&nbsp;</span></p>

{{{id=36|
for c in Compositions(4):
    F[c]
///
F[1, 1, 1, 1]
F[1, 1, 2]
F[1, 2, 1]
F[1, 3]
F[2, 1, 1]
F[2, 2]
F[3, 1]
F[4]
}}}

{{{id=40|

///
}}}

{{{id=37|

///
}}}

{{{id=42|

///
}}}

{{{id=41|
M.dual()
///
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
}}}

{{{id=43|

///
}}}

{{{id=44|
NCSF = NonCommutativeSymmetricFunctions(QQ)
///
}}}

{{{id=45|
NCSF.inject_shorthands()
///
Injecting S as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
Injecting R as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Ribbon basis
/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/categories/sets_cat.py:1225: RuntimeWarning: redefining global value `R`
  inject_variable(shorthand, realization)
Injecting L as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Elementary basis
Injecting Phi as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Phi basis
Injecting Psi as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
Injecting nM as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Monomial basis
Injecting I as shorthand for Non-Commutative Symmetric Functions over the Rational Field in the Immaculate basis
/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/categories/sets_cat.py:1225: RuntimeWarning: redefining global value `I`
  inject_variable(shorthand, realization)
}}}

{{{id=46|
S
///
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
}}}

{{{id=47|
NCSF.complete()
///
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
}}}

{{{id=48|
NCSF.Complete()
///
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
}}}

{{{id=49|
NCSF.S()
///
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
}}}

{{{id=51|

///
}}}

{{{id=50|

///
}}}

{{{id=52|
x = M[1,2]
///
}}}

{{{id=53|
x.expand(4)
///
x0*x1^2 + x0*x2^2 + x1*x2^2 + x0*x3^2 + x1*x3^2 + x2*x3^2
}}}

{{{id=54|
show(_)
///
<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}x_{0} x_{1}^{2} + x_{0} x_{2}^{2} + x_{1} x_{2}^{2} + x_{0} x_{3}^{2} + x_{1} x_{3}^{2} + x_{2} x_{3}^{2}</script></html>
}}}

{{{id=59|

///
}}}

{{{id=55|
LazyPowerSeriesRing(QQ)
///
Lazy Power Series Ring over Rational Field
}}}

{{{id=56|
PowerSeriesRing(QQ, 'x')
///
Power Series Ring in x over Rational Field
}}}

{{{id=57|

///
}}}

{{{id=58|
var('x')
///
x
}}}

{{{id=60|
f = 1/(1-x)
///
}}}

{{{id=61|
f.taylor(x, 0, 10)
///
x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
}}}

{{{id=63|

///
}}}

{{{id=62|

///
}}}

{{{id=64|
LPR = LazyPowerSeriesRing(QQ)
///
}}}

{{{id=65|
PSR = PowerSeriesRing(QQ, 'x', default_prec=10)
///
}}}

{{{id=66|
f = PSR(1)/PSR(1-x)
///
}}}

{{{id=67|
f
///
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)
}}}

{{{id=68|

///
}}}

{{{id=69|

///
}}}

{{{id=70|
M.pl
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_171.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TS5wbA=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmpoqPmGT/___code___.py", line 2, in <module>
    exec compile(u'M.pl' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "parent.pyx", line 620, in sage.structure.parent.Parent.__getattr__ (sage/structure/parent.c:6027)
  File "misc.pyx", line 200, in sage.structure.misc.getattr_from_other_class (sage/structure/misc.c:1406)
AttributeError: 'QuasiSymmetricFunctions.Monomial_with_category' object has no attribute 'pl'
}}}

{{{id=71|
f = h[2]
///
}}}

{{{id=72|
f.plethysm(h[2])
///
h[2, 2] - h[3, 1] + h[4]
}}}

{{{id=73|
x = M[2,1]
///
}}}

{{{id=74|
x.plethysm?
///
<html><!--notruncate-->
No object 'x.plethysm' currently defined.
</html>
}}}

{{{id=75|
f.plethysm(x)
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_187.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Zi5wbGV0aHlzbSh4KQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmpXY8Ocn/___code___.py", line 2, in <module>
    exec compile(u'f.plethysm(x)' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/sfa.py", line 1852, in plethysm
    raise TypeError, "only know how to compute plethysms between symmetric functions"
TypeError: only know how to compute plethysms between symmetric functions
}}}

{{{id=76|

///
}}}

{{{id=77|

///
}}}

{{{id=78|
s
///
Symmetric Functions over Rational Field in the Schur basis
}}}

{{{id=80|
Sym = SymmetricFunctions(QQ)
Sym.hall_littlewood()
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_208.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("U3ltLmhhbGxfbGl0dGxld29vZCgp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmpQDyZo2/___code___.py", line 2, in <module>
    exec compile(u'Sym.hall_littlewood()' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/sf.py", line 915, in hall_littlewood
    return hall_littlewood.HallLittlewood(self, t=t)
  File "classcall_metaclass.pyx", line 279, in sage.misc.classcall_metaclass.ClasscallMetaclass.__call__ (sage/misc/classcall_metaclass.c:932)
  File "cachefunc.pyx", line 569, in sage.misc.cachefunc.CachedFunction.__call__ (sage/misc/cachefunc.c:2868)
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/structure/unique_representation.py", line 463, in __classcall__
    instance = typecall(cls, *args, **options)
  File "classcall_metaclass.pyx", line 467, in sage.misc.classcall_metaclass.typecall (sage/misc/classcall_metaclass.c:1294)
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/hall_littlewood.py", line 92, in __init__
    raise ValueError, "parameter t must be in the base ring"
ValueError: parameter t must be in the base ring
}}}

{{{id=81|
Sym = SymmetricFunctions(QQ['t'])
H = Sym.hall_littlewood()
///
}}}

{{{id=82|
HP = H.P()
///
}}}

{{{id=83|
x = HP.an_element()
///
}}}

{{{id=84|
x
///
HLP[] + 2*HLP[1] + 3*HLP[2]
}}}

{{{id=85|
s(x)
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_224.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cyh4KQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/tmp/tmpBQURNz/___code___.py", line 2, in <module>
    exec compile(u's(x)' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "parent.pyx", line 804, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7305)
  File "coerce_maps.pyx", line 82, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3583)
  File "coerce_maps.pyx", line 77, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3485)
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/classical.py", line 206, in _element_constructor_
    return self(sx)
  File "parent.pyx", line 804, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7305)
  File "coerce_maps.pyx", line 82, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3583)
  File "coerce_maps.pyx", line 77, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3485)
  File "/home/saliola/Applications/sage-5.6/local/lib/python2.7/site-packages/sage/combinat/sf/classical.py", line 161, in _element_constructor_
    return eclass(self, dict([ (e1,R(e2)) for e1,e2 in x._monomial_coefficients.items()]))
  File "parent.pyx", line 804, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7305)
  File "polynomial_element.pyx", line 6554, in sage.rings.polynomial.polynomial_element.ConstantPolynomialSection._call_ (sage/rings/polynomial/polynomial_element.c:47620)
TypeError: not a constant polynomial
}}}

{{{id=86|
s = Sym.schur()
///
}}}

{{{id=87|
s
///
Symmetric Functions over Univariate Polynomial Ring in t over Rational Field in the Schur basis
}}}

{{{id=88|
s(x)
///
s[] + 2*s[1] - 3*t*s[1, 1] + 3*s[2]
}}}

{{{id=89|

///
}}}

{{{id=90|

///
}}}

{{{id=91|
Semigroups()
///
Category of semigroups
}}}

{{{id=92|
Semigroups().example()
///
An example of a semigroup: the left zero semigroup
}}}

{{{id=93|
QSym.category()
///
Join of Category of graded hopf algebras over Rational Field and Category of monoids with realizations and Category of coalgebras over Rational Field with realizations
}}}

{{{id=94|
for c in QSym.__class__.mro():
    print c
///
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions_with_category'>
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions'>
<class 'sage.structure.unique_representation.UniqueRepresentation'>
<type 'sage.structure.parent.Parent'>
<type 'sage.structure.category_object.CategoryObject'>
<type 'sage.structure.sage_object.SageObject'>
<class 'sage.categories.category.JoinCategory.parent_class'>
<class 'sage.categories.graded_hopf_algebras.GradedHopfAlgebras.parent_class'>
<class 'sage.categories.graded_bialgebras.GradedBialgebras.parent_class'>
<class 'sage.categories.graded_algebras.GradedAlgebras.parent_class'>
<class 'sage.categories.graded_coalgebras.GradedCoalgebras.parent_class'>
<class 'sage.categories.graded_modules.GradedModules.parent_class'>
<class 'sage.categories.hopf_algebras.HopfAlgebras.parent_class'>
<class 'sage.categories.bialgebras.Bialgebras.parent_class'>
<class 'sage.categories.algebras.Algebras.parent_class'>
<class 'sage.categories.rings.Rings.parent_class'>
<class 'sage.categories.rngs.Rngs.parent_class'>
<class 'sage.categories.monoids.Monoids.WithRealizations.parent_class'>
<class 'sage.categories.coalgebras.Coalgebras.WithRealizations.parent_class'>
<class 'sage.categories.coalgebras.Coalgebras.parent_class'>
<class 'sage.categories.vector_spaces.VectorSpaces.parent_class'>
<class 'sage.categories.modules.Modules.parent_class'>
<class 'sage.categories.bimodules.Bimodules.parent_class'>
<class 'sage.categories.left_modules.LeftModules.parent_class'>
<class 'sage.categories.right_modules.RightModules.parent_class'>
<class 'sage.categories.commutative_additive_groups.CommutativeAdditiveGroups.parent_class'>
<class 'sage.categories.semirings.Semirings.parent_class'>
<class 'sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.WithRealizations.parent_class'>
<class 'sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.parent_class'>
<class 'sage.categories.commutative_additive_semigroups.CommutativeAdditiveSemigroups.parent_class'>
<class 'sage.categories.additive_magmas.AdditiveMagmas.parent_class'>
<class 'sage.categories.monoids.Monoids.parent_class'>
<class 'sage.categories.semigroups.Semigroups.parent_class'>
<class 'sage.categories.magmas.Magmas.parent_class'>
<class 'sage.categories.sets_cat.Sets.WithRealizations.parent_class'>
<class 'sage.categories.facade_sets.FacadeSets.parent_class'>
<class 'sage.categories.sets_cat.Sets.parent_class'>
<class 'sage.categories.category.SetsWithPartialMaps.parent_class'>
<class 'sage.categories.objects.Objects.parent_class'>
<type 'object'>
}}}

{{{id=95|
for c in QuasiSymmetricFunctions(ZZ).__class__.mro():
    print c
///
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions_with_category'>
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions'>
<class 'sage.structure.unique_representation.UniqueRepresentation'>
<type 'sage.structure.parent.Parent'>
<type 'sage.structure.category_object.CategoryObject'>
<type 'sage.structure.sage_object.SageObject'>
<class 'sage.categories.category.JoinCategory.parent_class'>
<class 'sage.categories.graded_hopf_algebras.GradedHopfAlgebras.parent_class'>
<class 'sage.categories.graded_bialgebras.GradedBialgebras.parent_class'>
<class 'sage.categories.graded_algebras.GradedAlgebras.parent_class'>
<class 'sage.categories.graded_coalgebras.GradedCoalgebras.parent_class'>
<class 'sage.categories.graded_modules.GradedModules.parent_class'>
<class 'sage.categories.hopf_algebras.HopfAlgebras.parent_class'>
<class 'sage.categories.bialgebras.Bialgebras.parent_class'>
<class 'sage.categories.algebras.Algebras.parent_class'>
<class 'sage.categories.rings.Rings.parent_class'>
<class 'sage.categories.rngs.Rngs.parent_class'>
<class 'sage.categories.monoids.Monoids.WithRealizations.parent_class'>
<class 'sage.categories.coalgebras.Coalgebras.WithRealizations.parent_class'>
<class 'sage.categories.coalgebras.Coalgebras.parent_class'>
<class 'sage.categories.modules.Modules.parent_class'>
<class 'sage.categories.bimodules.Bimodules.parent_class'>
<class 'sage.categories.left_modules.LeftModules.parent_class'>
<class 'sage.categories.right_modules.RightModules.parent_class'>
<class 'sage.categories.commutative_additive_groups.CommutativeAdditiveGroups.parent_class'>
<class 'sage.categories.semirings.Semirings.parent_class'>
<class 'sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.WithRealizations.parent_class'>
<class 'sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.parent_class'>
<class 'sage.categories.commutative_additive_semigroups.CommutativeAdditiveSemigroups.parent_class'>
<class 'sage.categories.additive_magmas.AdditiveMagmas.parent_class'>
<class 'sage.categories.monoids.Monoids.parent_class'>
<class 'sage.categories.semigroups.Semigroups.parent_class'>
<class 'sage.categories.magmas.Magmas.parent_class'>
<class 'sage.categories.sets_cat.Sets.WithRealizations.parent_class'>
<class 'sage.categories.facade_sets.FacadeSets.parent_class'>
<class 'sage.categories.sets_cat.Sets.parent_class'>
<class 'sage.categories.category.SetsWithPartialMaps.parent_class'>
<class 'sage.categories.objects.Objects.parent_class'>
<type 'object'>
}}}

{{{id=96|
for c in QSym.an_element().__class__.mro():
    print c
///
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Monomial_with_category.element_class'>
<class 'sage.combinat.ncsf_qsym.qsym.Monomial.Element'>
<class 'sage.combinat.free_module.CombinatorialFreeModuleElement'>
<type 'sage.structure.element.Element'>
<type 'sage.structure.sage_object.SageObject'>
<class 'sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Bases.element_class'>
<class 'sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF.element_class'>
<class 'sage.categories.category.WithRealizations.ParentMethods.Realizations.element_class'>
<class 'sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis.element_class'>
<class 'sage.categories.graded_bialgebras_with_basis.GradedBialgebrasWithBasis.element_class'>
<class 'sage.categories.graded_algebras_with_basis.GradedAlgebrasWithBasis.element_class'>
<class 'sage.categories.graded_coalgebras_with_basis.GradedCoalgebrasWithBasis.element_class'>
<class 'sage.categories.graded_modules_with_basis.GradedModulesWithBasis.element_class'>
<class 'sage.categories.category.JoinCategory.element_class'>
<class 'sage.categories.graded_hopf_algebras.GradedHopfAlgebras.element_class'>
<class 'sage.categories.graded_bialgebras.GradedBialgebras.element_class'>
<class 'sage.categories.graded_algebras.GradedAlgebras.element_class'>
<class 'sage.categories.graded_coalgebras.GradedCoalgebras.element_class'>
<class 'sage.categories.graded_modules.GradedModules.element_class'>
<class 'sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis.element_class'>
<class 'sage.categories.bialgebras_with_basis.BialgebrasWithBasis.element_class'>
<class 'sage.categories.algebras_with_basis.AlgebrasWithBasis.element_class'>
<class 'sage.categories.coalgebras_with_basis.CoalgebrasWithBasis.element_class'>
<class 'sage.categories.modules_with_basis.ModulesWithBasis.element_class'>
<class 'sage.categories.category.HopfAlgebras.Realizations.element_class'>
<class 'sage.categories.hopf_algebras.HopfAlgebras.element_class'>
<class 'sage.categories.bialgebras.Bialgebras.element_class'>
<class 'sage.categories.algebras.Algebras.element_class'>
<class 'sage.categories.commutative_rings.CommutativeRings.element_class'>
<class 'sage.categories.rings.Rings.element_class'>
<class 'sage.categories.rngs.Rngs.element_class'>
<class 'sage.categories.category.Magmas.Realizations.element_class'>
<class 'sage.categories.category.Coalgebras.Realizations.element_class'>
<class 'sage.categories.coalgebras.Coalgebras.element_class'>
<class 'sage.categories.vector_spaces.VectorSpaces.element_class'>
<class 'sage.categories.modules.Modules.element_class'>
<class 'sage.categories.bimodules.Bimodules.element_class'>
<class 'sage.categories.left_modules.LeftModules.element_class'>
<class 'sage.categories.right_modules.RightModules.element_class'>
<class 'sage.categories.commutative_additive_groups.CommutativeAdditiveGroups.element_class'>
<class 'sage.categories.category.Semirings.element_class'>
<class 'sage.categories.commutative_additive_monoids.CommutativeAdditiveMonoids.element_class'>
<class 'sage.categories.commutative_additive_semigroups.CommutativeAdditiveSemigroups.element_class'>
<class 'sage.categories.additive_magmas.AdditiveMagmas.element_class'>
<class 'sage.categories.monoids.Monoids.element_class'>
<class 'sage.categories.semigroups.Semigroups.element_class'>
<class 'sage.categories.magmas.Magmas.element_class'>
<class 'sage.categories.category.Sets.Realizations.element_class'>
<class 'sage.categories.sets_cat.Sets.element_class'>
<class 'sage.categories.category.SetsWithPartialMaps.element_class'>
<class 'sage.categories.objects.Objects.element_class'>
<type 'object'>
}}}

{{{id=97|

///
}}}