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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
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,X1∪ ...∪ Xn) :=
(1)/(|G|)∑g∈ G ∏
i=1n(∏
j=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, edges, and faces. When denoting the families of
indeterminates for these actions by the following symbols
vi, ei, and
fi we compute:
|
Z3(R)=(1)/(60)( 24
v512e518f12f56+
20
v320e330f12f310+
15
v230e12e244f216+
v160e190f132).. |
and
|
Z3(S)=(1)/(2)Z3(R)+
(1)/(120)( 24
v106e109f2f103+
20
v610e615f2f65+
15
v14v228e18e241
f18f212+
v230e245f216)... |
Using these 3-dimensional cycle indices we can compute the
number of essentially different simultaneous colourings of
all vertices, edges and faces with
k1,k2,k3 colours by
replacing each variable vi by
k1, ei by k2
and so on. For k1= ...=k3 =2 the
number of R-different colourings is
|
102 166 369 391 059 257 223 889 803 047 960 653 063 858 332 471 656 448.. |
whereas the number of S-different colourings is
|
51 083 184 695 529 628 611 945 218 457 262 717 577 369 212 017 967 104... |
harald.fripertinger@kfunigraz.ac.at
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A
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