Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

TL;DR

This paper develops a polynomial-based method to enumerate and classify self-orthogonal partial Latin rectangles, providing explicit counts and formulas for small sizes, enhancing combinatorial understanding of these objects.

Contribution

It introduces a novel algebraic approach using Gröbner bases to explicitly enumerate and classify self-orthogonal partial Latin rectangles, including formulas for small parameters.

Findings

Cardinality of self-orthogonal partial Latin rectangles for small sizes computed

Explicit formulas for the number of such rectangles with r ≤ 3

Distribution of partial Latin rectangles based on size obtained

Abstract

The current paper deals with the enumeration and classification of the set $\mathcal{SOR}_{r,n}$ of self-orthogonal $r\times r$ partial Latin rectangles based on $n$ symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Gr\"obner basis and Hilbert series can be computed to determine explicitly the set $\mathcal{SOR}_{r,n}$. In particular, the cardinality of this set is shown for $r\leq 4$ and $n\leq 9$ and several formulas on the cardinality of $\mathcal{SOR}_{r,n}$ are exposed, for $r\leq 3$. The distribution of $r\times s$ partial Latin rectangles based on $n$ symbols according to their size is also obtained, for all $r,s,n\leq 4$.