Counting Nonattacking Chess Piece Placements: Bishops and Anassas

TL;DR

This paper develops formulas and recurrences for counting nonattacking arrangements of bishops and a new piece called the anassa on chessboards, providing explicit expressions and simplifying existing formulas.

Contribution

It introduces a general counting formula for the anassa and simplifies known formulas for bishops, expanding the understanding of nonattacking piece placements.

Findings

Derived recurrences and closed-form expressions for bishops and anassa.

Established explicit formulas for quasi-polynomial coefficients.

Simplified existing expressions for bishop placements.

Abstract

We derive recurrences and closed-form expressions for counting nonattacking placements of two types of chess pieces with unbounded straight-line moves, namely the bishop (two diagonal moves) and the anassa (one horizontal or vertical move and one diagonal move), placed on a standard square chessboard. Additionally, we obtain explicit expressions for the corresponding quasi-polynomial coefficients. The recurrences are derived by analyzing how nonattacking configurations attack a specific subset of board squares, employing a bijective argument to establish the relations. The main results are simplifications of known expressions for the bishop and a general counting formula for the anassa.