[GAP Forum] cycle index and Polya enumeration

[Hebert Pérez-Rosés]

Yes, it's very helpful. Thanks.


2011/2/2 Alexander Konovalov <alexander.konovalov at gmail.com>

> Dear Hebert, dear GAP Forum,
>
> GAP 4.4.12 already has the non-documented function "CycleIndex"
> which you may use. It will become documented in the next release
> of GAP 4.5 as follows:
>
> CycleIndex( g, Omega[, act] )
> CycleIndex( G, Omega[, act] )
>
> The cycle index of a permutation g acting on Omega is defined as
>
> z(g) = s_1^{c_1} s_2^{c_2} cdots s_n^{c_n}
>
> where c_k is the number of k-cycles in the cycle decomposition
> of g and the s_i are indeterminates.
>
> The cycle index of a group G is defined as
>
> Z(G) = ( sum_{g in G} z(g) ) / |G| .
>
> The indeterminates used by CycleIndex are the indeterminates 1 to n
> over the rationals.
>
> gap> g:=TransitiveGroup(6,8);
> S_4(6c) = 1/2[2^3]S(3)
> gap> CycleIndex(g);
> 1/24*x_1^6+1/8*x_1^2*x_2^2+1/4*x_1^2*x_4+1/4*x_2^3+1/3*x_3^2
>
> Hope this helps,
> Alexander
>
>
> On 19 Dec 2010, at 12:22, Hebert Pérez-Rosés wrote:
>
> > Dear all,
> >
> > Does anybody have a GAP function to compute the cycle index of a
> permutation
> > group, and perform Polya enumeration?
> >
> > Best regards,
> >
> > Hebert Perez-Roses
> > The University of Newcastle, Australia
>