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Collections of cells, binomial ideals and combinatorics

Carmelo Cisto, Rizwan Jahangir, Francesco Navarra

TL;DR

The paper presents PolyominoIdeals, a Macaulay2 package that unifies combinatorial and algebraic perspectives on collections of cells and polyominoes. It provides robust encodings (via lower-left corners) and a suite of combinatorial tools (connectivity, convexity, holes, random generation) alongside rich algebraic constructs such as inner and adjacent $2$-minor ideals, toric and lattice representations, and Gröbner basis analyses. A central theme is the link between combinatorial structures and algebraic invariants, exemplified by the relationship between rook polynomials and Hilbert-Poincaré series, including switching variants. The framework enables explicit computation of ideals, their toric counterparts, and associated invariants, with practical implications for combinatorial commutative algebra and computational algebraic geometry.

Abstract

In this paper we provide a description of the package \textit{PolyominoIdeals} for \textit{Macaulay2} that allows to deal with collections of cells, polyominoes and related binomial ideals.

Collections of cells, binomial ideals and combinatorics

TL;DR

The paper presents PolyominoIdeals, a Macaulay2 package that unifies combinatorial and algebraic perspectives on collections of cells and polyominoes. It provides robust encodings (via lower-left corners) and a suite of combinatorial tools (connectivity, convexity, holes, random generation) alongside rich algebraic constructs such as inner and adjacent -minor ideals, toric and lattice representations, and Gröbner basis analyses. A central theme is the link between combinatorial structures and algebraic invariants, exemplified by the relationship between rook polynomials and Hilbert-Poincaré series, including switching variants. The framework enables explicit computation of ideals, their toric counterparts, and associated invariants, with practical implications for combinatorial commutative algebra and computational algebraic geometry.

Abstract

In this paper we provide a description of the package \textit{PolyominoIdeals} for \textit{Macaulay2} that allows to deal with collections of cells, polyominoes and related binomial ideals.
Paper Structure (3 sections, 3 theorems, 16 equations, 6 figures)

This paper contains 3 sections, 3 theorems, 16 equations, 6 figures.

Table of Contents

  1. Collections of cells, polyominoes and some combinatorial functions
  2. Commutative algebra related to collections of cells
  3. Rook polynomial theory and a connection with the Hilbert-Poincaré series

Key Result

Theorem 2.1

CNU1 Let $\mathcal{Q}$ be a simple and weakly connected collection of cells. Then $I_{\mathcal{Q}}=J_{\mathcal{Q}}$.

Figures (6)

  • Figure 1: A collection of cells $\mathcal{Q}$.
  • Figure 2: A weakly connected collection of cells with two connected components on the left and a polyomino on the right.
  • Figure 3: Some collections of cells.
  • Figure 4: Positions of two rooks in a polyomino.
  • Figure 5: Square tetromino and two equivalent $2$-rook configurations.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Example 2.6