Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Switching rook polynomial of collections of cells

Francesco Navarra, Ayesha Asloob Qureshi, Giancarlo Rinaldo

TL;DR

This work establishes and tests a deep link between algebraic invariants of polyomino/collection ideals and combinatorial rook polynomials. It extends the switching rook polynomial to general collections of cells, provides a computational framework to compute these polynomials, and proves that for convex collections under specific Gröbner-basis conditions, the switching rook polynomial coincides with the h-polynomial of the associated coordinate ring while the rook number matches the Castelnuovo–Mumford regularity. Computational evidence supports the conjectured equality across broad classes (up to rank 10 collections and rank 12 polyominoes), and a general convex-case theorem is proved, enabling algebraic methods to enumerate rook configurations. The results offer a cohesive algebraic approach to classical rook-placement problems and enrich the interplay between combinatorics and commutative algebra.

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.

Switching rook polynomial of collections of cells

TL;DR

This work establishes and tests a deep link between algebraic invariants of polyomino/collection ideals and combinatorial rook polynomials. It extends the switching rook polynomial to general collections of cells, provides a computational framework to compute these polynomials, and proves that for convex collections under specific Gröbner-basis conditions, the switching rook polynomial coincides with the h-polynomial of the associated coordinate ring while the rook number matches the Castelnuovo–Mumford regularity. Computational evidence supports the conjectured equality across broad classes (up to rank 10 collections and rank 12 polyominoes), and a general convex-case theorem is proved, enabling algebraic methods to enumerate rook configurations. The results offer a cohesive algebraic approach to classical rook-placement problems and enrich the interplay between combinatorics and commutative algebra.

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 -minors. We design an algorithm to compute the switching rook polynomial of a collection of cells and show that it coincides with the -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.

Paper Structure

This paper contains 5 sections, 12 theorems, 27 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Required terminologies related to collection of cells
  2. Ideals of 2-minors Associated to Collections of Cells
  3. Switching rook polynomial of a collection of cells and a computational approach
  4. Quadratic Gröbner basis of collections of cells
  5. Hilbert series of convex collection of cells with quadratic Gröbner basis

Key Result

Proposition 2.2

Let $\mathcal{B}_{m,n}$ be a rectangular polyomino of width $m$ and height $n$. Then, for every $0 \leq k \leq \min\{m,n\}$, we have

Figures (8)

  • Figure 1: A polyomino, a weakly connected collection of cells with two connected components, and a collection of cells with three weakly connected components.
  • Figure 2: From left to right: a Ferrer diagram, a stack, a parallelogram and a directed convex.
  • Figure 3: Two examples of $5$-rook configurations in $\mathcal{P}$.
  • Figure 4: Conditions for a quadratic Gröbner basis with respect to $<_{\mathrm{rev}}$ and $<_{\operatorname{lex}}$.
  • Figure 5: Two weakly connected convex collections of cells and a weakly connected, simple and non-convex collection of cells, whose connected components are convex.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Conjecture 1
  • Remark 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Conjecture 2.4
  • Theorem 3.1: Q
  • Remark 3.2
  • ...and 19 more