
Math.Combinatorics.Species.CycleIndex 




Description 
An instance of Species for cycle index series. For details on
cycle index series, see "Combinatorial Species and TreeLike
Structures", chapter 1.


Synopsis 



Documentation 


Convert a cycle index series to an exponential generating
function: F(x) = Z_F(x,0,0,0,...).



Convert a cycle index series to an ordinary generating function:
F~(x) = Z_F(x,x^2,x^3,...).



Extract a particular coefficient from a cycle index series.



Compute fix F[n], i.e. the number of Fstructures fixed by a
permutation with cycle type n, given the cycle index series Z_F.
In particular, fix F[n] = aut(n) * zCoeff Z_F n.


Miscellaneous



aut js is is the number of automorphisms of a permutation with
cycle type js (i.e. a permutation which has n cycles of size
i for each (i,n) in js). Another way to look at it is that
there are n!/aut js permutations on n elements with cycle type
js. The result type is a FactoredRational.T.



Generate all partitions of an integer. In particular, if p is
an element of the list output by intPartitions n, then sum
. map (uncurry (*)) $ p == n. The result type is [CycleType]
since each integer partition of n corresponds to the cycle type
of a permutation on n elements.
The partitions are generated in an order corresponding to
the Ord instance for Monomial.



cyclePower s n computes the cycle type of sigma^n, where sigma
is any permutation of cycle type s.
In particular, if s = (s_1, s_2, s_3, ...) (i.e. sigma has s_1
fixed points, s_2 2cycles, ... s_k kcycles), then
sigma^n_j = sum_{j*gcd(n,k) = k} gcd(n,k)*s_k


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