Switching rook polynomial of collections of cells

TL;DR

This paper investigates the relationship between rook placements on collections of cells and algebraic properties of ideals, introducing an algorithm for computing the switching rook polynomial and exploring its algebraic significance.

Contribution

It introduces a new algorithm for the switching rook polynomial and establishes a conjectured correspondence with the $h$-polynomial of associated coordinate rings, proven for specific convex collections.

Findings

Algorithm computes switching rook polynomial for collections of cells.

Conjecture that switching rook polynomial equals the $h$-polynomial of the coordinate ring.

Proven the conjecture for certain convex collections.

Abstract

We explore the novel connection between rook placements on collections of cells, also known as pruned chessboards, and the algebraic properties of ideals generated by $2$-minors. We design an algorithm to compute the switching rook polynomial of a collection of cells and show that it coincides with the $h$-polynomial of the associated coordinate ring for all collections up to rank 10 and polyominoes up to rank 12. Motivated by this evidence, we conjecture that the correspondence holds in general, and we prove it for certain convex collections of cells by algebraic tools.