Enumerating partial Latin rectangles

TL;DR

This paper introduces computational methods to enumerate partial Latin rectangles, deriving formulas and exact counts for various parameters, and classifying them up to isomorphism using algebraic geometry and combinatorial techniques.

Contribution

It provides new formulas and enumeration techniques for partial Latin rectangles, extending previous methods to larger sizes and classifying them via algebraic and combinatorial approaches.

Findings

Exact formulas for partial Latin rectangles with up to 13 non-empty cells.

Enumeration of partial Latin rectangles for sizes up to 8x8.

Classification of rectangles into isomorphism and isotopism classes.

Abstract

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leq 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leq s \leq n \leq 7$, and all $r \leq s \leq 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leq s \leq n \leq 6$. Numerical results have been cross-checked where possible.