The 3-dimensional Cycle Index of the Leapfrog of a Polyhedron

Harald Fripertinger¹

November 18, 1996

Abstract

Relations between the 3-dimensional cycle index of the point group of a trivalent polyhedron or of a deltahedron on the one hand and of its leapfrog on the other hand are described.

The Leapfrog transformation is a method first invented for the construction of a fullerene C_3n from a parent C_n having the same as or even a bigger symmetry group than C_n. It was introduced by P.W. Fowler in his papers [2]FowlerSteer87. (Molecules in the form of 3-connected polyhedral cages with exactly 12 pentagonal and all the other hexagonal faces solely built from carbon atoms are called fullerenes. Fullerenes C_n can be constructed for n=20 and for all even n≥ 24. They have n vertices (i.e. C-atoms), 3n/2 edges and (n-20)/2 hexagonal faces. The most important member of the family of the fullerenes is C₆₀.)

In general the leapfrog transformation can be defined for any polyhedron P as capping all the faces of P and switching to the dual of the result. The leapfrog L(P) is always a trivalent polyhedron having 2e_P vertices, v_P+f_P faces and 3e_P edges, where v_P, f_P and e_P are the numbers of vertices, faces and edges of the parent P. When starting from a trivalent parent, the leapfrog has always 3v_P vertices.

In [6] it is described how the symmetry group of a fullerene C_n (especially for n=20, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50 52, 54, 56, 58, 60, 70, 80 and 140) acts on its sets of vertices, faces and edges. Then general techniques from the theory of enumeration under finite group actions [7] are applied for determining the number of isomers of these molecules, or in other words for counting all the essentially different colourings of C_n. (Two colourings are called essentially different if they lie in different orbits of the symmetry group of C_n acting on the set of all colourings of C_n.) Especially a 3-dimensional cycle index for the simultaneous action of the symmetry group on the sets of vertices, edges and faces of C_n is presented.

Whenever a group G is acting on sets X₁,..., X_n then G acts in a natural way on the disjoint union

X:= n
∪
i=1
X_i.

The n-dimensional cycle index which uses for each set X_i a separate family of indeterminates x_i,1,x_i,2,... is given by

Z_n(G,X₁∪ ...∪ X_n) := ^._. 1

|G|
^._.
∑
g∈ G ^._. n
∏
i=1 (^._. |X_i|
∏
j=1 x_i,j^a_i,j(g)),^._.

where (a_i,1(g),...,a_{i,|X_i|}(g)) is the cycle type of the permutation corresponding to g and to the action of g on X_i. (I.e. the induced permutation on X_i decomposes into a_i,j disjoint cycles of length j for j=1,...,|X_i|.) For the action on the sets of vertices, edges and faces we usually denote the indeterminates by v_i, e_i and f_i. Using the n-dimensional cycle index it is possible to determine the number of essentially different simultaneous colourings of X₁∪ ...∪ X_n as described in [6].

For instance the 3-dimensional cycle index for the action of the octahedral group O_h acting on the cube is given by

Z₃(O_h,cube)=^._. 1

48
( v₁⁸e₁¹²f₁⁶ + 8v₁²v₃²e₃⁴f₃² + 6v₂⁴e₁²e₂⁵f₂³ + 3v₂⁴e₂⁶f₁²f₂² + 6v₄²e₄³f₁²f₄+ 6v₁⁴v₂²e₁²e₂⁵f₁²f₂² + v₂⁴e₂⁶f₂³ + 3v₂⁴e₁⁴e₂⁴f₁⁴f₂ + 8v₂v₆e₆²f₆ + 6v₄²e₄³f₂f₄). ^._.

These cycle indices are the basic tools for applying Pólya-theory [8] to the isomer count. It was already mentioned in [6] that the cycle types for the action on the set of faces of the leapfrog can easily be obtained from the 3-dimensional cycle index of the action on the parent. But for the actions on the sets of vertices and edges of the leapfrog we did not give satisfying methods.

Using the notation of spherical shell techniques the permutation representations for the actions on the sets of vertices, edges or faces of a polyhedron correspond to the so called σ representations. In [3][4] it is shown how the σ representations Γ_σ(v,L), Γ_σ(e,L) and Γ_σ(f,L) for the actions on the components of the leapfrog L=L(P) of an arbitrary polyhedron P are related to the σ representations Γ_σ(v,P), Γ_σ(e,P) and Γ_σ(f,P) corresponding to the parent:

Γ_σ(f,L)=Γ_σ(v,P)+Γ_σ(f,P) ^._.

Γ_σ(v,L)=Γ_σ(e,P)+Γ_σ(f,P)+ Γ_σ(v,P) × Γ_ε-(Γ₀+Γ_ε) ^._.

Γ_σ(e,L)=Γ_σ(f,L) × Γ_T -(Γ_T+Γ_R) ^._.

where Γ₀ is the totally symmetric representation with character χ₀(g)=1 for all g. The character of the antisymmetric representation Γ_ε is +1 for all proper rotations and -1 for all improper rotations. Γ_T (or Γ_xyz) is the translational representation, which is the representation of a set of cartesian unit vectors at the origin, and Γ_R=Γ_T × Γ_ε is the rotational representation.

These formulae can be rewritten in order to get the permutation characters for all g in the symmetry group G of P by

χ_f,L(g)=χ_v,P(g)+χ_f,P(g) ^._.

χ_v,L(g)=χ_e,P(g)+χ_f,P(g)+χ_v,P(g) χ_ε(g)-(1+ χ_ε(g)) ^._.

χ_e,L(g)=χ_f,L(g)χ_T(g)-(χ_T(g)+χ_R(g)) ^._.

So far the permutation characters for the action on the components of the leapfrog are expressed in the permutation characters for the action on the components of the parent and in χ_ε and χ_T. Since usually the cycle indices both of the group of all symmetries and of the subgroup of all rotational symmetries of the parent are known we can assume that the antisymmetric character is known. Only for applying formula (*) we furthermore have to compute the translational character. In some cases however all the necessary information for computing χ_T is given by the 3-dimensional cycle index for the action on the parent P.

For instance if P is a trivalent polyhedron (see [1]), then

Γ_σ(e,P)=Γ_σ(f,P) × Γ_T -(Γ_T+Γ_R). ^._.

Combining (*) and (*) formula (*) can be written as

Γ_σ(e,L)=(Γ_σ(v,P)+Γ_σ(f,P)) × Γ_T -(Γ_T+Γ_R) =Γ_σ(v,P) × Γ_T +Γ_σ(e,P).^._.

From [1] we deduce that

Γ_σ(v,P) × Γ_T = Γ_||(e,P) + Γ_σ(e,P) ^._.

and

Γ_||(e,P)=(Γ_σ(f,P)-Γ₀) × Γ_ε+ (Γ_σ(v,P)-Γ₀), ^._.

where Γ_|| is the parallel representation. So finally (*) can be replaced by

Γ_σ(e,L)=(Γ_σ(f,P)-Γ₀) × Γ_ε +(Γ_σ(v,P)-Γ₀) +Γ_σ(e,P)+Γ_σ(e,P) ^._.

and the permutation character χ_e,L(g) can be computed as

χ_e,L(g)=2χ_e,P(g)+(χ_f,P(g)-1) χ_ε(g)+ (χ_v,P(g)-1). ^._.

If P is a deltahedron, which is the dual of a trivalent polyhedron, then (*) can be replaced by

χ_e,L(g)=2χ_e,P(g)+(χ_v,P(g)-1) χ_ε(g)+ (χ_f,P(g)-1). ^._.

Using standard methods [7] the cycle type of g∈ G can be computed from the permutation character of g and vice versa by

a_k(g)=^._.
∑
d | k μ(k/d)a₁(g^d) a₁(g^k)=^._.
∑
d | k a_d(g), ^._.

where μ is the classical Möbius function.

Given a trivalent polyhedron or a deltahedron P with symmetry group G and subgroup H of rotational symmetries. Then the 3-dimensional cycle indices for the actions of G and H on the leapfrog L(P) can be computed from the 3-dimensional cycle indices for the actions on the parent P as described above. It is worth to mention once more that no further group characters must be computed. In other words the 3-dimensional cycle indices for the action on the parent provide all the necessary information.

For example the cycle index for the leapfrog of the cube can be computed as:

Z₃(O_h,L)=^._. 1

48
( v₁²⁴e₁³⁶f₁¹⁴ + 8v₃⁸e₃¹²f₁²f₃⁴ + 6v₂¹²e₁²e₂¹⁷f₂⁷ + 3v₂¹²e₂¹⁸f₁²f₂⁶ + 6v₄⁶e₄⁹f₁²f₄³+ 6v₂¹²e₁²e₂¹⁷f₁⁶f₂⁴ + v₂¹²e₂¹⁸f₂⁷ + 3v₁⁸v₂⁸e₁¹²e₂¹²f₁⁴f₂⁵ + 8v₆⁴e₆⁶f₂f₆² + 6v₄⁶e₄⁹f₂f₄³). ^._.

In order to give another example we realize that C₆₀ is the leapfrog of C₂₀. They both are of icosahedral symmetry I_h, the subgroup of all proper rotations will be denoted by I. In [6] the following 3-dimensional cycle indices for the actions on the components of C₂₀ can be found.

Z₃(I,C₂₀)=^._. 1

60
( v₁²⁰e₁³⁰f₁¹² + 20v₁²v₃⁶e₃¹⁰f₃⁴ + 15v₂¹⁰e₁²e₂¹⁴f₂⁶ + 24v₅⁴e₅⁶f₁²f₅²) ^._.

Z₃(I_h,C₂₀)= ^._. 1

2
Z₃(I,C₂₀) + ^._. 1

120
( v₂¹⁰e₂¹⁵f₂⁶ + 20v₂v₆³e₆⁵f₆² + 15v₁⁴v₂⁸e₁⁴e₂¹³f₁⁴f₂⁴ + 24v₁₀²e₁₀³f₂f₁₀).^._.

Applying (*), (*), (*) and (*) we compute:

Z₃(I,C₆₀)=^._. 1

60
( v₁⁶⁰e₁⁹⁰f₁³²+ 20 v₃²⁰e₃³⁰f₁²f₃¹⁰+ 15 v₂³⁰e₁²e₂⁴⁴f₂¹⁶+ 24 v₅¹²e₅¹⁸f₁²f₅⁶ ) ^._.

and

Z₃(I_h,C₆₀)= ^._. 1

2
Z₃(I,C₆₀) + ^._. 1

120
( v₂³⁰e₂⁴⁵f₂¹⁶+ 20 v₆¹⁰e₆¹⁵f₂f₆⁵+ 15 v₁⁴v₂²⁸e₁⁸e₂⁴¹ f₁⁸f₂¹²+ 24 v₁₀⁶e₁₀⁹f₂f₁₀³ ). ^._.

Iterating the leapfrog method once more we derive the 3-dimensional cycle index of C₁₈₀ as

Z₃(I,C₁₈₀)=^._. 1

60
( v₁¹⁸⁰e₁²⁷⁰f₁⁹²+ 20 v₃⁶⁰e₃⁹⁰f₁²f₃³⁰+ 15 v₂⁹⁰e₁²e₂¹³⁴f₂⁴⁶+ 24 v₅³⁶e₅⁵⁴f₁²f₅¹⁸ ) ^._.

and

Z₃(I_h,C₁₈₀)= ^._. 1

2
Z₃(I,C₁₈₀) + ^._. 1

120
( v₂⁹⁰e₂¹³⁵f₂⁴⁶+ 20 v₆³⁰e₆⁴⁵f₂f₆¹⁵+ 15 v₁¹²v₂⁸⁴e₁¹²e₂¹²⁹ f₁¹²f₂⁴⁰+ 24 v₁₀¹⁸e₁₀²⁷f₂f₁₀⁹ ). ^._.

In order to compute the number of essentially different colourings of C_3n it is necessary to compute the 3-dimensional cycle index for the action on C_3n and apply the methods described in [6]. Only for the determination of the number of different colourings of the faces of C_3n with k colours the 3-dimensional cycle index of C_n will do the job in the following way. Replace all the indeterminates in this cycle index corresponding to the actions on the sets of vertices and faces of C_n by k and all the indeterminates corresponding to the action on the set of edges by 1, then the expansion of this cycle index gives the number of different colourings of the faces of C_3n. For example the number of essentially different simultaneous colourings of C₂₀ with 2 colours for the vertices, 1 colour for the edges and 2 colours for the faces is computed as

Z₃(C₂₀,I_h, v_i=2, e_i=1, f_i=2)=35 931 952,^._.

which is the number of different colourings of the faces of C₆₀ with 2 colours (cf. [6]). It should be mentioned that this number is not the product of the numbers of different colourings of the vertices and faces of C₂₀ with 2 colours. (These two numbers are given as 9 436 and 82 respectively.)

Acknowledgement: The author wants to express his thanks to Prof. A. Kerber and Prof. P. Fowler for their support and guidance during the preparation of this article.

harald.fripertinger "at" uni-graz.at, May 10, 2016

The 3-dimensional Cycle Index of the Leapfrog of a Polyhedron

Harald Fripertinger1

November 18, 1996

Abstract

Harald Fripertinger¹