Some cycle indices

# 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].
• The symmetry group of the fullerene C60
• The action on the set of vertices
• The action on the set of edges
• The action on the set of faces

• harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

 Some cycle indices