Arrangements with given number of neighbours

Abstract

A method previously employed to find the number p(N, n, X ) of ways of arranging n B particles on N points on a straight line while producing X pairs of BB nearest neighbours is extended to find the number of ways of arranging given numbers of A, B, C, ... particles on N points lying in a number of parallel planes containing in each a finite number of points while producing given numbers of various kinds of neighbours (such as the nearest neighbours, the next nearest neighbours, etc.) formed by different pairs of particles. In the former case, the method gives us the limiting value ofas N approaches infinity, and in the present case the limiting value of N -1 times the logarithm of our required number as the number of parallel planes approaches infinity. The number of ways of arranging n B particles on two rows of points while producing X pairs of BB nearest neighbours is investigated in detail, and the results plotted. Finally, it is pointed out that if we can evaluate certain determinants of infinite number of rows and columns, we can find the number of ways of arrangements when the N points are arranged on a crystalline lattice.