Some cycle indices
For applying Pólya theory to the
combinatorics of the fullerene C60 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 g1·(g2·x)=(g1g2)·x and 1·x=x for
all g1,g2ÎG 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
x1,x2,... over the set of rationals given by
Z(G,X):=(1)/(|G|)ågÎG
Õi=1|X|xiai(bar ( g)),
where bar ( g) is the permutation representation of g and
(a1(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
