Level and pseudo-Gorenstein path polyominoes

TL;DR

This paper classifies and analyzes level and pseudo-Gorenstein path polyominoes, providing a comprehensive classification and computing properties for polyominoes with small rank, enhancing understanding of their algebraic and combinatorial structure.

Contribution

It offers the first complete classification of level and pseudo-Gorenstein path polyominoes and computes their regularity relative to rank for simple thin cases.

Findings

Classified all level and pseudo-Gorenstein path polyominoes.

Computed all such polyominoes with rank ≤ 10.

Established the relationship between regularity and rank for these polyominoes.

Abstract

We classify path polyominoes which are level and pseudo-Gorenstein. Moreover, we compute all level and pseudo-Gorenstein simple thin polyominoes with rank less than or equal to 10. We also compute the regularity of the pseudo-Gorenstein simple thin polyominoes in relation to their rank.