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

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.