 |  |  | A multi-dimensional cycle index |
A multi-dimensional cycle index
Whenever a group G is acting on sets
X1,..., Xn then G acts in a
natural way on the disjoint union
X:=Èi=1n Xi.
Replacing in such a situation the cycle
index of G acting on X by a
so-called n-dimensional cycle
index we get more information about
the permutation representation of G.
The n-dimensional cycle index which
uses for each set Xi a separate family of indeterminates
xi,1,xi,2,... is given by
Zn(G,X1È...ÈXn):=(1)/(|G|)ågÎG
Õi=1n(Õj=1|Xi|xi,jai,j(g)),
where (ai,1(g),...,ai,|Xi|(g)) is the cycle type of the
permutation corresponding to g acting on Xi.
Returning to the fullerene C60 the groups
R and S are acting on the
disjoint union of the sets of all
vertices, pentagonal edges, hexagonal edges,
pentagonal faces, hexagonal faces and diagonals.
When denoting the families
of indeterminates for these actions by the following symbols
vi, ei, Ei, fi, Fi and di we compute:
Z6(R)=(1)/(60)(
24 v512e512E56f12f52F54d56+
20 v320e320E310f34F12F36d310+
15 v230e230E12E214f26F210d12d214+
v160e160E130f112F120d130)
and
Z6(S)=(1)/(2)Z6(R)+ (1)/(120)(
24 v106e106E103f2f10F102d56+
20 v610e610E65f62F2F63d310+
15 v14v228e14e228E14E213
f14f24F14F28d12d214+
v230e230E215f26F210d130).
From these cycle indices we deduce that the action
on the sets of vertices
and pentagonal edges have the same cycle type.
The variables Ei, fi and
Fi determine the cycle index of the
symmetry group of the icosahedron
acting on its set of edges, vertices or faces.
Using these 6-dimensional cycle indices we can compute the number
of different
simultaneous colourings of all vertices, pentagonal and
hexagonal edges, pentagonal and hexagonal faces and diagonals with
k1,k2,...,k6 colours by replacing each variable vi by k1,
ei by k2 and so on.
For k1= ...=k6 =2 the number of
R-different colourings is
109700303821413736143664612170571163303931905179435773317873664
whereas the number of
S-different colourings is
54850151910706868071832306128208569015853860985570356157743104.
This substitution into an n-dimensional cycle index can be computed
using the SYMMETRICA routine polya_multi_const_sub(a,b,c).
harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001
| | | A multi-dimensional cycle index |