Permutation/2 problem

Figure 2 40. To illustrate the isomorphism problem, phenylalanine is simplified to a core without representing the substituents. Then every core atom is numbered arbitrarily (first line). On this basis, the substituents of the molecule can be permuted without changing the constitution (second line). Each permutation can be represented through a permutation group (third line). Thus the first line of the mapping characterizes the numbering of the atoms before changing the numbering, and the second line characterizes the numbering afterwards. In the initial structure (/) the two lines are identical. Then, for example, the substituent number 6 takes the place of substituent number 4 in the second permutation (P2), when compared with the reference molecule.

The previous analysis has shown that the properties of unidirectional fibre composites are highly anisotropic. To alleviate this problem, it is common to build up laminates consisting of stacks of unidirectional lamina arranged at different orientations. Clearly many permutations are possible in terms of the numbers of layers (or plies) and the relative orientation of the fibres in each... 

Cayley s extensive computations have been checked and, where necessary, adjusted. Real progress has been achieved by two American chemists, Henze and Blair Not only did the two authors expand Cayley s computations, but they also improved the method and introduced more classes into the compound. Lunn and Senior , on the other hand, discovered independently of Cayley s problems that certain numbers of isomers are closely related to permutation groups. In the present paper, I will extend Cayley s problems in various ways, expose their relationship with the theory of permutation groups and with certain functional equations, and determine the asymptotic values of the numbers in question. The results are described in the next four chapters. More detailed summaries of these chapters are given below. Some of the results presented here in detail have been outlined before ... 

The combinatorial problem on permutation groups stands out for its generality and the simplicity of the solution. The following... 

The problem of which the example in Sec. 2 represents a very special case can be stated as follows Given the collection of figures [ ], the permutation group H and the content (k,H,m), determine the number of nonequivalent configurations of content (k,Jl,m) with... 

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. 

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... 

The numbering of the edges and the proper permutation groups describe the problem completely and, therefore, it can be treated in purely combinatorial terms. 

To make use of the group G we need some way of summarizing those properties of the group that are relevant to the problem. This was provided by Polya in the form of the "cycle index". It is well known that a permutation can be expressed as a product of disjoint... 

A theorem which, at first sight, does not seem to be very closely related to Polya s Theorem, but which in fact has much affinity with it, is the superposition theorem that appeared in my doctoral thesis [ReaR58] and later in [ReaR59,60]. The general problem to which it applies is the following. Consider an ordered set of k permutation groups of degree , say G. G. . and the set of all A -ads... 

This provides an inductive, and a constructive, proof of the possibility of a triangular factorization of the specified form, provided only certain submatrices are nonsingular. For suppose first, that Au is a scalar, A12 a row vector, and A21 a column vector, and let Ln = 1. Then i u = A1U B12 — A12, and L2l and A22 axe uniquely defined, provided only Au = 0. But Au can be made 0, at least after certain row permutations have been made. Hence the problem of factoring the matrix A of order n, has been reduced to the factorization of the matrix A22 of order n — 1. 

It is not possible to describe and provide fine-tuned controls for every potential waterside or steamside problem that may arise, in view of the countless permutations of boiler plants and specific operating conditions that exist around the world. But many problems are inevitably more common than others, and some of these are described in the various sections of the next four chapters, together with some practical notes on their control. Although many of the issues described have been neatly compartmentalized for the sake of simplicity, this generally is not the case in practice and waterside or steamside problems should always be investigated in the context of the overall boiler plant. 

As can be seen from the various bullet points listed at the beginning of this chapter, the permutations of potential problems by industry, process, equipment type, and specific area are extremely wide. Consequently, to preempt such problems developing, or at least to con-... 

After you have done this, we can start focusing on the main techniques for analyzing problems that display permutations that you have never seen. That is what the next section is all about. 

Representation. The solution space is composed of discrete combinatorial alternatives of batch production schedules. For example, in the permutation flowshop problem, where the batches are assumed to be executed in the same order on each unit, there are A number of solutions, where N is the number of batches. We must find a way to compactly represent this solution space, in such a way that significant portions of the space can be characterized with respect to our objective as either poor or good without explicitly enumerating them. 

Lagweg, BJ., Lenstra, J.K., and Rinnooy Kan, A.H.G., A general bounding scheme for the permutation flow-shop problem. Oper. Res. 26(1) (1978). 

The vast number of thermodynamically possible reactions obtained by permuting oxidants and reductants within the scope of this review present major problems of classification and selection. To only a limited extent is the modernity or detail of a paper indicative of its relevance, some of the definitive papers having been published before 1950. Discussion has been concentrated, therefore, at points where a kinetic investigation of a reaction has resulted in a real advance in our understanding both of its mechanism and of those of related reactions, and work which has been more of a confirmatory nature will not receive comparable consideration. Detailed reference to products, spectra, etc. will be made only when the kinetics produce real ambiguities. 

Applying the permutation operator P12 is therefore equivalent to interchanging rows of the determinant in Eq. (2.15). Having devised a method for constructing many-electron wavefunctions as a product of MOs, the final problem concerns the form of the many-electron Hamiltonian which contains terms describing the interaction of a given electron with (a) the fixed atomic nuclei and (b) the remaining (N— 1) electrons. The first step is therefore to decompose H(l, 2, 3,..., N) into a sum of operators Hj and H2, where ... 

---