On the number of linear partitions of the (m, n)-grid

Abstract

A relationship between linear partitions and minimal pairs of a finite point set in the plane was established in [2]. This relationship is used here for counting the number of linear partitions of the set of points of the (m, n)-grid, a rectangular part of the infinite grid. In order to optimize this counting, an O(mn) algorithm is introduced for traversing all those pairs (i, j) of mutually simple natural numbers i and j, such that 1 ≤i≤m, 1≤j≤n.