Powers enumerated

In theory and upon paper, this apparatus of powers, seems amply sufficient for all general purposes. In several material instances, they exceed the powers enumerated in the articles of confederation. The Amphyc-tions had in their hands the superstition of the times, one of the principal engines by which government was then maintained they had declared authority to use coercion against refractory cities, and were bound by oath to exert this authority on the necessary occasions. 

The power group enumeration theorem has been put to a great number of uses, especially in graph theoretical and chemical enumeration. One simple example will illustrate the value of the theorem. 

Let us consider again, and now solve, the necklace problem that was mentioned in the discussion of De Bruijn s Theorem, namely, to enumerate necklaces of six beads of two colors, red and green, where the two colors can be interchanged. We shall ask only for the total number, and can therefore use a simpler (unweighted) form of the power group enumeration theorem which gives as the required number the expression... 

The use of cycle index sums, mentioned above, is a powerful technique for enumeration, and a comparatively recent development of Polya theory. It will be instructive to consider a simple example... 

Powerful though Polya s Theorem undoubtedly is, it is not difficult to formulate problems which, though superficially very similar to the general Polya-type problem, cannot be solved by use of the theorem. We have already seen how some problems of this kind led to generalizations of Polya s Theorem, such as de Bruijn s Theorem, the power group enumeration theorem, and the superposition theorem, but even these theorems have their limitations. For some problems not amenable to solution by any of these results, it is sufficient to fall back on the result from which they all stem, namely Burnside s Lemma. 

HarF66 Harary, F., Palmer, E. M The power group enumeration theorem. J. Comb. Theory 1 (1966) 157-173. 

The explosives enumerated in Table 155 are very powerful type D oxyliquits. Their power may be reduced by the addition of inert cooling substances, e.g. sodium chloride, kieselguhr etc. An explosive is then obtained with a performance similar to... 

Applying this criterion to. a lattice model, v corresponds roughly to the volume of a unit cell, and / to a lattice spacing. Hence the values of n 10 for exact enumeration quoted in Section IV seem quite reasonable. Certainly there is no support for the claim by Flory and Fisk31 that the 6/5 power law is attained only for n > 10.6... 

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