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
such that g1⋅ (g2⋅
x)=(g1g2)⋅ x and 1⋅ x=x for all
g1,g2∈ G and x∈ X. The
orbit of x∈ X is the set G(x) of all elements of the
form g⋅ x for g∈ G. 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|
|
.. |
∑
g∈ G |
.. |
|X|
∏
i=1 |
xiai(g),.. |
where g is the
permutation representation of g and
(a1(g),...,
a|X|(g))
is the cycle type of the permutation g. For more details about cycle
indices (and about combinatorics via finite group actions in
general) see [18].
harald.fripertinger "at" uni-graz.at, May 10,
2016
