Invariant point configurations?a combinatorial approach

Abstract

A combinatorial and algebraic approach has been applied to the problem of determining the number of distinct configurations of univariant reaction lines about an invariant point, NC, in nondegenerate n-component systems. The resulting expression is $$NC = [1/(4n + 8)]\left( {\sum {[2^{d/2} (\phi (2n + 4)/d]) + (n + 2) 2^{\{ (n + 2)/2\} } } } \right) - 1$$ where the summation is taken over all values of d which divide (2n + 4)evenly but which do not divide (n + 2)evenly, [(n + 2)/2]is the smallest integer greater than or equal to (n + 2)/2,and o[(2n + 4)/d]is the number of integers less than (2n + 4)/d whose only factor is common with (2n + 4)/d is 1.This concise expression is derived through the application of Burnside's lemma, which relates the number of equivalence classes into which a set S is divided by an equivalence relation induced by a permutation group of S to the number of elements of S left invariant by the members of the permutation group. In the derivation of the above expression, the set S is taken to be the set of all possible configurations and orientations of univariant lines about an invariant point, the permutation group is taken to be the set of rigid-body symmetries of S, and the equivalence classes are composed of the different orientations of each configuration. Although the methods used to obtain the above expression are probably unfamiliar to most geologists, they are standard mathematical techniques and represent just one application of these tools to problems of geologic interest.