Abstract
In this paper we are concerned with a problem in the Pólya theory of enumeration. Our main result is the determination of the cycle index of a group SH which acts on a set of equivalence classes VN/H. (Let V and N be finite, non-empty sets and let S and H be permutation groups on V and N respectively; VN is the set of injective mappings of N into V; elements f, g ∈ VN are H-equivalent if there exists θ ∈ H such that fθ = g, and VN/H is the set of H-equivalence classes. Any σ ∈ S induces, in a natural way, a permutation τH(σ) of VN/H, and SH is the group of all τH(σ)'s.) This result is used to enumerate classes of structures such as linear graphs, directed graphs, and generalized graphs. A typical consequence of our results is the determination of the cycle structure of the permutation of the subsets of a finite set induced by a permutation of the set.
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