Using Words to Construct and Enumerate Maximum Nonattacking Chessboard Arrangements

Words are ubiquitous objects in combinatorics that can be studied in their own right or be used to represent sets of other combinatorial objects. Additionally, mathematical questions related to chess, its pieces, its board, and the many extensions and generalizations, have been posed for hundreds of years. In particular, chess pieces can have move sets beyond those in the established game of chess. Here we use words to describe nonattacking arrangements of chess pieces on a rectangular \(2\times 2n\) chessboard. We then extend the notion of words on a single line to a matrix of letters, focusing on pieces who can move with attacks from a king’s move set. Bijections between matrices of letters in small alphabets and nonattacking maximum arrangements of pieces on rectangular chessboards are used to enumerate such arrangements.