Index cycle

Formula ( ) is the base for asymptotic computations of and and (1") lends itself to generalizations. (Indeed, in the general term of the series on the right hand side of (1") we recognize the cycle index of the symmetric group of n elements.)... 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

In an earlier paper [Polya, 4], I used "Symmetrieformel" instead of "cycle index". 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

We note that (1.12) relates to the cycle index (1.16) like (1.15) (taking (1.12) and (1.14) into account) to (1.17). The following definitions allow us to state the rules on the construction of the generating functions in a unified way. To introduce the functions /(x), f(x,y) into the cycle index means putting... 

Theorem. The generating function for the configurations [] which are nonequivalent with respect to H is obtained by substituting the generating function of [4>] in the cycle index of U. 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

The cycle index of E, is obvious, for Z, and D, it is easily derived. For Z and summation extends over all divisors of s. The special cases for the smallest values of s are reflected in the above table, since... 

The problem of Sec. 12 can be stated in this special situation as follows Let It be an arbitrary permutation group of degree 5 and /cj, /cj,. .., denote n non-negative integers whose sum is s. How many nonequivalent ways modulo H are there to place /Cj balls of the first, balls of the second,. .., k balls of the n-th color in 5 slots According to Sec. 16 the solution is established by introducing m cycle index of H and expanding the... 

Relations Between Cycle Index and Permutation Group... 

The property of the cycle index given in Sec. 16 uniquely determines the cycle index. More accurately Let /(xj,. .) denote the... 

The main theorem, stated in Sec. 16 and proved in Sec. 19, combined with the proposition of Sec. 25 yields the following proposition Tvfo permutation groups are combinatorially equivalent if and only if they have the same cycle index. 

Referring to the definition (1.5) of the cycle index we find further two permutation groups are combinatorially equivalent if and only if there exists a unique correspondence between the permutations of the two groups such that corresponding permutations have the same type of cycle decomposition. 

We discuss some cases in which the cycle index of a group composed of several groups can be constructed from the cycle indices of the given groups in a transparent way. 

For the spatial congruence, that is, with group and its cycle index (cf. Sec. 21), we find... 

A third permutation group of the graph of cyclopropane obtains if the regular prism discussed as a model for (a) is subjected to rotations as well as to reflections which leave it invariant. The six vertices are thus subject to a permutation group of order 12. We call it the extended group of the stereoformula. Its cycle index is... 

We note that chemical substitution of a radical into a basic compound corresponds (in the sense of the main theorem of Chapter 1) to the algebraic substitution of the generating function into the cycle index of the group of the basic compound. 

The example demonstrates that the concepts in chemistry rely heavily on notions from group theory, specifically the concept, introduced in Sec. 11, of the equivalence of configurations with respect to a permutation group. The cycle index and the main theorem of Sec. 16 play a role. 

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