| | | Orbit-enumeration in SYMMETRICA |
Let G be a permutation group of X (if necessary take the homomorphic image of G under f:G -> SX). The cycle index of G acting on X is the following polynomial Z(G,X), which is a polynomial in the indeterminates x1,x2,...,x | X | over Q, defined by
Z(G,X):=(1)/(|G|)ågÎGÕi=1|X|xiai(g),where (a1(g),...,a|X| (g)) is the cycle type of the permutation gÎG. This means, g decomposes into ai(g) disjoint cycles of length i for i=1,...,|X|.
At first we will present some cycle index formulae for natural actions of cyclic, dihedral, symmetric groups etc. Using multi-dimensional cycle indices the cycle indices of the symmetry groups of the five platonic solids, and of some fullerenes can be described. Some cycle index routines for linear affine and projective groups are also implemented. From a given cycle index the cycle indices of the induced actions on the sets of all k-tuples, or all k-subsets can be computed. Furthermore there are some interesting group actions of the direct product and of the wreath product of two permutation groups discussed.
| | | Orbit-enumeration in SYMMETRICA |