本文整理匯總了Python中sage.combinat.sf.sf.SymmetricFunctions.term方法的典型用法代碼示例。如果您正苦於以下問題：Python SymmetricFunctions.term方法的具體用法？Python SymmetricFunctions.term怎麽用？Python SymmetricFunctions.term使用的例子？那麽, 這裏精選的方法代碼示例或許可以為您提供幫助。您也可以進一步了解該方法所在類sage.combinat.sf.sf.SymmetricFunctions的用法示例。
在下文中一共展示了SymmetricFunctions.term方法的1個代碼示例，這些例子默認根據受歡迎程度排序。您可以為喜歡或者感覺有用的代碼點讚，您的評價將有助於係統推薦出更棒的Python代碼示例。
示例1: cycle_index
# 需要導入模塊: from sage.combinat.sf.sf import SymmetricFunctions [as 別名]
# 或者: from sage.combinat.sf.sf.SymmetricFunctions import term [as 別名]
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]
#.........這裏部分代碼省略.........