53.ランプパターンの転移問題 : 単因子論の一応用

Abstract

An mn lamp pattern is a distribution of the on-off states of the mn lamps arranged in an mn rectangular array. If one touches one of the lamps, then the on-off status of that lamp, and of the vertically-adjacent or horizontally-adjacent lamps will all be reversed. This is a basic transition, and these transitions applied successively define an equivalence relation among the set of the mn lamp patterns. This paper is concerned with determination of the number of the equivalence classes of the mn lamp patterns. It is shown that the class number is given by 2^d, with the degree d of the polynomial G.C.D. (det (xI_n-A_n), det ((x-1)I_m-A_m)), where I_n is the unit matrix and A_n is the incidence matrix of a basic transition, containing 1 on the two lines parallel and adjacent to the main diagonal and 0 elsewhere