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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 φ: 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|
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∑
g∈ G
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|X|
∏
i=1
|
xiai(g), |
where (a1(g),...,a|X| (g)) is the cycle type
of the permutation g∈ G. 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.
harald.fripertinger "at" uni-graz.at, May 26,
2011
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Orbit-enumeration in SYMMETRICA |
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