Minimal Free Resolutions of $2\times n$ Domino Tilings

TL;DR

This paper studies the algebraic structure of a monomial ideal linked to domino tilings of a 2 by n rectangle, revealing characteristic-independent Betti numbers and explicit formulas for projective dimension and regularity.

Contribution

It introduces a new monomial ideal for domino tilings and explicitly determines its minimal free resolution, including projective dimension and regularity, using algebraic and combinatorial methods.

Findings

Betti numbers are characteristic-independent.

Explicit formulas for projective dimension and regularity.

Minimal free resolution constructed for the ideal.

Abstract

We introduce a squarefree monomial ideal associated to the set of domino tilings of a $2\times n$ rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo-Mumford regularity of the ideal.