A computational algebraic geometry approach to classify partial Latin rectangles

TL;DR

This paper applies computational algebraic geometry techniques to classify and count partial Latin rectangles, deriving explicit formulas and discovering new configurations for structures up to size six and point rank eight.

Contribution

It introduces a novel algebraic geometry approach to classify partial Latin rectangles and determine explicit counts, including new configurations at higher ranks.

Findings

Explicit formulas for partial Latin rectangles up to size six.

Classification of seminet structures up to point rank eight.

Discovery of two new configurations of point rank eight.

Abstract

This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying $r\times s$ partial Latin rectangles based on $n$ symbols of a given size, shape, type or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all $r,s,n\leq 6$. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of size up to six. We focus then on the study of non-compressible regular partial Latin squares and their equivalent incidence structure called seminet, whose distribution into main classes is explicitly determined for point rank up to eight. We prove in particular the existence of two new configurations of point rank eight.