def cycle_index(self, parent = None):
r"""
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the p basis)
Returns the *cycle index* of `G`, which is a gadget counting
the elements of `G` by cycle type, averaged over the group:
.. math::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Polya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
TODO: add some simple examples of Polya enumeration, once it
will be easy to expand symmetric functions on any alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # optional - database_gap # long time
... G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should have a ``term`` and ``sum`` method::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
AssertionError: `parent` should be (or behave as) a free module with basis indexed by partitions
REFERENCES:
.. [Ker1991] A. Kerber. Algebraic combinatorics via finite group actions, 2.2 p. 70.
BI-Wissenschaftsverlag, Mannheim, 1991.
AUTHORS:
- Nicolas Borie and Nicolas M. Thiery
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.combinat.permutation import Permutation
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
else:
assert hasattr(parent, "term") and hasattr(parent, "sum"), \
"`parent` should be (or behave as) a free module with basis indexed by partitions"
base_ring = parent.base_ring()
# TODO: use self.conjugacy_classes() once available
from sage.interfaces.gap import gap
CC = ([Permutation(self(C.Representative())).cycle_type(), base_ring(C.Size())] for C in gap(self).ConjugacyClasses())
return parent.sum( parent.term( partition, coeff ) for (partition, coeff) in CC)/self.cardinality()
def cycle_index(self, parent=None):
r"""
Return the *cycle index* of ``self``.
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the `p` basis)
The *cycle index* of a permutation group `G`
(:wikipedia:`Cycle_index`) is a gadget counting the
elements of `G` by cycle type, averaged over the group:
.. MATH::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Pólya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
.. TODO::
Add some simple examples of Pólya enumeration, once
it will be easy to expand symmetric functions on any
alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # long time
....: G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
Permutation groups with arbitrary domains are supported
(see :trac:`22765`)::
sage: G = PermutationGroup([['b','c','a']], domain=['a','b','c'])
sage: G.cycle_index()
1/3*p[1, 1, 1] + 2/3*p[3]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should be a module with basis indexed by partitions::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
ValueError: `parent` should be a module with basis indexed by partitions
REFERENCES:
- [Ke1991]_
AUTHORS:
- Nicolas Borie and Nicolas M. Thiéry
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.categories.modules import Modules
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
elif not parent in Modules.WithBasis:
raise ValueError(
"`parent` should be a module with basis indexed by partitions"
)
base_ring = parent.base_ring()
return parent.sum_of_terms(
[C.an_element().cycle_type(),
base_ring(C.cardinality())]
for C in self.conjugacy_classes()) / self.cardinality()
def cycle_index(self, parent = None):
r"""
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the p basis)
Returns the *cycle index* of `G`, which is a gadget counting
the elements of `G` by cycle type, averaged over the group:
.. math::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Polya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
TODO: add some simple examples of Polya enumeration, once it
will be easy to expand symmetric functions on any alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # optional - database_gap # long time
... G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should have a ``term`` and ``sum`` method::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
AssertionError: `parent` should be (or behave as) a free module with basis indexed by partitions
REFERENCES:
.. [Ker1991] A. Kerber. Algebraic combinatorics via finite group actions, 2.2 p. 70.
BI-Wissenschaftsverlag, Mannheim, 1991.
AUTHORS:
- Nicolas Borie and Nicolas M. Thiery
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.combinat.permutation import Permutation
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
else:
assert hasattr(parent, "term") and hasattr(parent, "sum"), \
"`parent` should be (or behave as) a free module with basis indexed by partitions"
base_ring = parent.base_ring()
# TODO: use self.conjugacy_classes() once available
from sage.interfaces.gap import gap
CC = ([Permutation(self(C.Representative())).cycle_type(), base_ring(C.Size())] for C in gap(self).ConjugacyClasses())
return parent.sum( parent.term( partition, coeff ) for (partition, coeff) in CC)/self.cardinality()
def cycle_index(self, parent = None):
r"""
Return the *cycle index* of ``self``.
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the `p` basis)
The *cycle index* of a permutation group `G`
(:wikipedia:`Cycle_index`) is a gadget counting the
elements of `G` by cycle type, averaged over the group:
.. MATH::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Pólya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
.. TODO::
Add some simple examples of Pólya enumeration, once
it will be easy to expand symmetric functions on any
alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # optional - database_gap # long time
....: G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
Permutation groups with arbitrary domains are supported
(see :trac:`22765`)::
sage: G = PermutationGroup([['b','c','a']], domain=['a','b','c'])
sage: G.cycle_index()
1/3*p[1, 1, 1] + 2/3*p[3]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should be a module with basis indexed by partitions::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
ValueError: `parent` should be a module with basis indexed by partitions
REFERENCES:
- [Ke1991]_
AUTHORS:
- Nicolas Borie and Nicolas M. Thiéry
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.categories.modules import Modules
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
elif not parent in Modules.WithBasis:
raise ValueError("`parent` should be a module with basis indexed by partitions")
base_ring = parent.base_ring()
return parent.sum_of_terms([C.an_element().cycle_type(), base_ring(C.cardinality())]
for C in self.conjugacy_classes()
) / self.cardinality()