# Some cycle indices

For applying Pólya theory to the
combinatorics of the fullerene C_{60} we
must determine the *cycle index*
of the *symmetry group* of the
truncated icosahedron.
Let *G* be a multiplicative group and let
*X* be a set then a *group
action* of *G* on *X* is given by a mapping
*G´X -> X, (g,x) -> g·x,*

such that *g*_{1}·(g_{2}·x)=(g_{1}g_{2})·x and *1·x=x* for
all *g*_{1},g_{2}ÎG and *xÎX*.
The *orbit* of *xÎX* is the set
*G(x)* of all elements of the form *g·x* for *gÎG*.
The cycle index of a finite group *G* acting on a finite set
*X* is a polynomial in indeterminates
*x*_{1},x_{2},... over the set of rationals given by
*Z(G,X):=***(**1**)/(**|G|**)**å_{gÎG}
Õ_{i=1}^{|X|}x_{i}^{ai(bar ( g))},

where *bar ( g)* is the *permutation representation* of *g* and
*(a*_{1}(bar (g)),..., a_{|X|}(bar (g)))
is the *cycle type* of the
permutation *bar (g)*.
For more details about cycle indices (and about combinatorics via
finite group actions in general) see [18].

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001