# A q-queens problem. VII. Combinatorial types of nonattacking chess riders

## TL;DR

This paper proves a conjecture about the number of combinatorial types of nonattacking configurations of identical chess riders with r moves on convex polygonal boards, showing it depends only on r.

## Contribution

It confirms a conjecture relating to the enumeration of combinatorial types of nonattacking chess riders with r moves, independent of specific move sets.

## Key findings

- Number of combinatorial types equals r(r^2+3r-1)/3 for three riders.
- The count is independent of the specific moves of the riders.
- The result applies to convex polygonal chessboards.

## Abstract

On a convex polygonal chessboard, the number of combinatorial types of nonattacking configuration of three identical chess riders with $r$ moves, such as queens, bishops, or nightriders, equals $r(r^2+3r-1)/3$, as conjectured by Chaiken, Hanusa, and Zaslavsky (2019). Similarly, for any number of identical 3-move riders the number of combinatorial types is independent of the actual moves.