# Introduction

In 1985 the first C_{60} clusters were produced and it was assumed
[21]
that its 60 carbon atoms are placed at the vertices of a *truncated
icosahedron* (which is commonly called a *foot ball* or *soccer ball*.)
The truncated icosahedron is one of the 14 *Archimedean solids*
already known to the ancient Greeks.
Five years later [20][24]
the original hypotheses on the structure of C_{60} could be confirmed.
A nice introduction into the theory of the fullerenes
providing historical and chemical background information
can be found in [10].
From the mathematical point of view in the present paper
we are interested in the
combinatorics of the symmetry group of the truncated icosahedron and
we will discuss some examples in how many ways the soccer ball
can be coloured in *essentially different* ways.
From the chemical point of view we will determine the number of all
possible placements of double bonds for the C_{60} fullerene.
Furthermore investigating the C_{60}H_{60}-molecule
(i.e. a hypothetic molecule where all the thirty double bonds
of the C_{60} are hydrated),
we will compute the numbers of all
*C*_{60}H_{k}Cl_{60-k} molecules.
Finally the implementation of these cycle index methods
in the computer algebra system SYMMETRICA [23]
will be discussed and some further cycle indices will be listed.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001