The Problem of the Kings

On a 2m 2n chessboard, the maximum number of nonattacking kings that can be placed is mn, since each 22 cell can have at most one king. Let f m (n) denote the number of ways that mn nonattacking kings can be placed on a 2m 2n chessboard. The purpose of this paper is to prove the following result. such that f m (n) = (c m n + d m)(m + 1) n + O(n m) (n ! 1): For every such placement of kings, the chessboard is naturally divided into mn 2 2 cells, each containing exactly one king. Let's say that a cell is of type 1 (resp. 2, 3, 4) if the king sits in its NW (resp. NE, SE, SW) corner. The arrangement of kings is then completely speciied by an m n matrix whose entries are 2 f1; 2; 3; 4g. For example, the array 1 4 1 1 2 3 3 2 2 3 4 4 4 3 3 (1) corresponds to a legal arrangement of kings on a 6 10 board, namely the following one. Conversely, a matrix such as (1) represents an allowable connguration of kings ii it satisses certain adjacency conditions, namely that none of the following two letter words is permitted: