Regularity and Gorenstein property of the $L$-convex Polyominoes

TL;DR

This paper investigates the algebraic properties of the coordinate ring of $L$-convex polyominoes, linking regularity to rook placements, characterizing Gorenstein cases, and computing Cohen--Macaulay types.

Contribution

It provides a new characterization of the Gorenstein property and regularity of $L$-convex polyominoes, connecting combinatorial and algebraic aspects.

Findings

Regularity is determined by the maximum number of rooks that can be placed.

Characterization of Gorenstein $L$-convex polyominoes.

Explicit computation of Cohen--Macaulay type based on covering rectangles.

Abstract

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it.