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 ¿ H such that f¿ = g, and VN/H is the set of H-equivalence classes. Any s S induces, in a natural way, a permutation tH(s) of VN/H, and SH is the group of all tH(s)'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.
| Original language | English |
|---|---|
| Pages (from-to) | 1-9 |
| Journal | Indagationes Mathematicae (Proceedings) |
| Volume | 72 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 1969 |