Hilbert series of Parallelogram Polyominoes

TL;DR

This paper explores the Hilbert series of parallelogram polyominoes, proposing a conjecture linking it to non-attacking rook arrangements, and confirms it for specific classes through computational and combinatorial methods.

Contribution

It introduces a conjecture relating Hilbert series to rook arrangements and proves it for parallelogram polyominoes using combinatorial and computational techniques.

Findings

Conjecture holds for all simple polyominoes up to rank 11.

Conjecture is proven for parallelogram polyominoes as planar distributive lattices.

Provides a combinatorial interpretation of Gorenstein properties.

Abstract

We present a conjecture about the reduced Hilbert series of the coordinate ring of a simple polyomino in terms of particular arrangements of non-attacking rooks that can be placed on the polyomino. By using a computational approach, we prove that the above conjecture holds for all simple polyominoes up to rank $11$. In addition, we prove that the conjecture holds true for the class of parallelogram polyominoes, by looking at those as simple planar distributive lattices. Finally, we give a combinatorial interpretation of the Gorensteinnes of parallelogram polyominoes.