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

Pantaree Thengarnanchai; Pawaton Kaemawichanurat; Watcharintorn Ruksasakchai; Natawat Klamsakul

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

Combinatorics 2024-07-11 v1

Authors: Pantaree Thengarnanchai , Pawaton Kaemawichanurat , Watcharintorn Ruksasakchai , Natawat Klamsakul

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.

Keywords

Latin rectangles chromatic polynomial mathematical induction coloring lattice partitions

Cite

@article{arxiv.2407.07378,
  title  = {A simple proof on the number of $(3 \times n)$-Latin rectangles based on a set of $\lambda$ elements},
  author = {Pantaree Thengarnanchai and Pawaton Kaemawichanurat and Watcharintorn Ruksasakchai and Natawat Klamsakul},
  journal= {arXiv preprint arXiv:2407.07378},
  year   = {2024}
}

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8 pages, 2 figures

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

Abstract

Keywords

Cite

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