A simple proof on the number of $(3 \times n)$-Latin rectangles based on a set of $\lambda$ elements

TL;DR

This paper provides a simplified proof for counting the number of $(3 imes n)$-Latin rectangles with entries from a set of $\lambda$ elements, improving upon previous methods that used complex combinatorial tools.

Contribution

The authors introduce a more straightforward proof leveraging induction and coloring techniques, avoiding the complex M"{o}bius inversion and lattice partition methods.

Findings

Simplified formula for the number of $(3 imes n)$-Latin rectangles

Reduction in proof complexity compared to previous approaches

Enhanced understanding of Latin rectangle enumeration

Abstract

In 1980, Athreya, Pranesachar and Singhi established the chromatic polynomial of $(3 \times n)$-Latin rectangles whose entries based on a set $\{1, 2, ..., \lambda\}$ in which $\lambda \geq n$. Their proof requires M\"{o}bius inversion formula and lattice partitions. In this paper, we present a simpler proof by using the idea of mathematical induction and appropriate coloring.