[GAP Forum] cycle index and Polya enumeration

[Alexander Konovalov]

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