Non-simple polyominoes of Kőnig type and their canonical module

Abstract

We study the Kőnig type property for non-simple polyominoes. We prove that, for closed path polyominoes, the polyomino ideals are of Kőnig 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.

Figures (21)

• Figure 1: A polyomino.
• Figure 2: An example of two closed paths.
• Figure 3: Changes of directions in a closed path.
• Figure 4: Some configurations in a closed path with zig-zag walks.
• Figure 5: Some configurations in a closed path with zig-zag walks.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4
• Theorem 3.5
• proof
• Corollary 3.6
• proof