| | | **Orbit-enumeration in SYMMETRICA** |

### Orbit-enumeration in SYMMETRICA

Let *G* be a permutation group of *X* (if necessary take the
homomorphic image of *G* under *f:G -> S*_{X}). The
*cycle index* of
*G* acting on *X* is the following polynomial *Z(G,X)*, which is a polynomial
in the indeterminates *x*_{1},x_{2},...,x_{ | X | } over **Q**, defined by

*Z(G,X):=***(**1**)/(**|G|**)**å_{gÎG}Õ_{i=1}^{|X|}x_{i}^{ai(g)},

where *(a*_{1}(g),...,a_{|X|} (g)) is the cycle type of the permutation
*gÎG*. This means, *g* decomposes into
*a*_{i}(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.

harald.fripertinger@kfunigraz.ac.at,

last changed: November 19, 2001

| | | **Orbit-enumeration in SYMMETRICA** |