Description
PolyominoIdeals is a package for making several computations with collections of cells, polyominoes and binomial ideals attached to them.
Collections of cells, and in particular polyominoes, are plane figures formed by joining unit squares edge to edge. The systematic study of polyominoes was initiated by Solomon W. Golomb in 1953 and further developed in his seminal monograph [G1994]. Beyond their combinatorial interest in problems such as tiling the plane, polyominoes also connect to several other fields, including theoretical computer science, statistical physics, and discrete geometry.
More recently, since 2012, polyominoes and, more generally, collections of cells have been investigated from an algebraic–combinatorial perspective. In [AAQ2012], Ayesha Asloob Qureshi established a link between combinatorial commutative algebra and collections of cells by associating to each collection the binomial ideal generated by its inner $2$-minors.
Consider the natural partial order on $\NN^2$ and let $a,b \in \N^2$ with $a\leq b$. The set $[a, b] = \{c \in \NN^2 : a \leq c \leq b\}$ is called an interval of $\NN^2$; moreover, if $b=a+(1,1)$, then $[a,b]$ is called a cell of $\NN^2$. An interval $C=[a, b]$, where $a = (i, j)$ and $b = (k, l)$, is said to be a proper interval if $i < k$ and $j < l$. The elements $a, b$ are said the diagonal corners of $C$ and $c = (k, j)$ and $d = (i, l)$ the anti-diagonal ones. If $C$ is a cell, then $V(C)=\{a,a+(1,1),a+(0,1),a+(1,0)\}$ is the set of the corners of $C$.
To a collection of cells $\mathcal{P}$ we associate the ideal $I_{\mathcal{P}}$ as follows. Let $K$ be a field and set $S = K[x_a : a\in V(\mathcal{P})]$, where $V(\mathcal{P})$ is the union of the vertex sets of the cells in $\mathcal{P}$. A proper interval $[a,b]$ is called an inner interval of $\mathcal{P}$ if every cell contained in $[a,b]$ belongs to $\mathcal{P}$. If $c$ and $d$ are the anti-diagonal corners of an inner interval $[a,b]$, then the binomial $f = x_a x_b - x_c x_d$ is called an inner $2$-minor of $\mathcal{P}$. If we restrict the generators to those corresponding only to the cells, we obtain the so-called adjacent $2$-minor ideals, introduced in [HH2012].
A collection $\mathcal{P}$ is a polyomino if for every two cells $A,B\in\mathcal{P}$ there exists a sequence $A=C_1,\dots,C_m=B$ of cells in $\mathcal{P}$ such that $C_i$ and $C_{i+1}$ share a common edge for all $i$. In this case $I_{\mathcal{P}}$ is called the polyomino ideal of $\mathcal{P}$.
The aim of this package is to provide several tools to help mathematicians in the study of collections of cells, polyominoes and the related binomial ideals. Every contribution is very welcome.
Literature
- [G1994] Polyominoes, Puzzles, Patterns, Problems, and Packagings(S.W. Golomb, 1994, Second edition, Princeton University Press).
- [HH2012] Ideals generated by adjacent 2-minors(J. Herzog, T. Hibi, 2012, J. Commut. Algebra).
- [AAQ2012] Ideals generated by 2-minors, collections of cells and stack polyominoes (A. A. Qureshi, 2012, J. Algebra).