Non-simple polyominoes of K\H{o}nig type and their canonical module

TL;DR

This paper investigates the K"onig type property in non-simple polyominoes, extending known results to closed path polyominoes and providing combinatorial insights into their algebraic structures.

Contribution

It proves that closed path polyomino ideals are of K"onig type and offers a combinatorial interpretation of the canonical module for certain classes.

Findings

Closed path polyomino ideals are of K"onig type

Canonical module has a combinatorial interpretation for circle closed path polyominoes

The coordinate ring is Cohen-Macaulay and level

Abstract

We study the K\H{o}nig type property for non-simple polyominoes. We prove that, for closed path polyominoes, the polyomino ideals are of K\H{o}nig type, extending the results of Herzog and Hibi for simple thin polyominoes. As an application of this result, we give a combinatorial interpretation for the canonical module of the coordinate ring of a sub-class of closed path polyominoes, namely circle closed path polyominoes. In this case, we compute also the Cohen-Macaulay type and we show that $K[\mathcal{P}]$ is a level ring.