Some counting problems Top Introduction 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,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 ∏ i=1|X|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 [13].
harald.fripertinger@kfunigraz.ac.at

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