   Colourings of the C_{60}H_{60}molecule 
Colourings of the C_{60}H_{60}molecule
The cycle indices for the action on the set of vertices
can be applied to investigate the C_{60} H_{60}molecule.
Consider a colouring of the 60 vertices C with 2 colours H or Cl.
In other words a colouring is a function f from the set of vertices
of the truncated icosahedron into the set {H,Cl} .
Then Pólyas theorem says that
the number of C_{60}H_{k}Cl_{60k} molecules is given
as the coefficient of x^{k} in the expansion of
the cycle index of the symmetry group R or S
acting on the set of vertices
when all the indeterminates x_{i} are replaced by 1+x^{i}.
(In mathematical terms we are speaking of weighted enumeration when
we define the weight of a colouring as a product weight.
Let w(H)=x and w(Cl)=1,
where x is an indeterminate over the set of rationals.
Then the product weight of the colouring f is defined to be
w(f):=Õ_{v}f(v),
where the product runs over all vertices v.)
In table you can find
the numbers of different molecules
C_{60}H_{k}Cl_{60k}
both for the symmetry groups R and S.
k  S  R 
0, 60  1  1 
1, 59  1  1 
2, 58  23  37 
3, 57  303  577 
4, 56  4190  8236 
5, 55  45718  91030 
6, 54  418470  835476 
7, 53  3.220218  6.436782 
8, 52  21.330558  42.650532 
9, 51  123.204921  246.386091 
10, 50  628.330629  1256.602779

Number of C_{60}H_{k}Cl_{60k} molecules
The substitution x_{i} > 1+x^{i} into the cycle index is implemented
in SYMMETRICA as well.
It is called polya_sub(a,b,c)
, where a
is the cycle index, b
is the number of
indeterminates x_{i} that should be replaced by 1+x^{i}
and c
is the result of the expansion of the cycle index.
More or less in the same way the numbers of
hetero fullerenes, these are fullerenes where some of the
carbon atoms are replaced by other atoms, can be computed from
the cycle index of the symmetry group acting on the set of
vertices.
For instance replacing some C atoms by B atoms the
numbers in table
can be interpreted as the numbers of
different C_{60k}B_{k} molecules.
harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001
   Colourings of the C_{60}H_{60}molecule 