The monochromatic column problem with a prime number of colors

Crowell, Loran; Szabo, Steve

doi:10.2140/involve.2019.12.1415

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The monochromatic column problem with a prime number of colors

Loran Crowell and Steve Szabo

Vol. 12 (2019), No. 8, 1415–1422

DOI: 10.2140/involve.2019.12.1415

Abstract

Let $p_{1}, \dots, p_{n}$ be a sequence of $n$ pairwise coprime positive integers, $P = p_{1} \dots p_{n}$ , and $0, \dots, m - 1$ be a sequence of $m$ different colors. Let $A$ be an $n \times m P$ matrix of colors in which row $i$ consists of blocks of $p_{i}$ consecutive entries of the same color with colors 0 through $m - 1$ repeated cyclically. The monochromatic column problem is to determine the number of columns of $A$ in which every entry is the same color. The solution for a prime number of colors is provided.

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Keywords

monochromatic column problem, Chinese remainder theorem, multiple sequence alignment problem

Mathematical Subject Classification 2010

Primary: 05A15, 11A07

Milestones

Received: 5 July 2019

Revised: 6 August 2019

Accepted: 12 August 2019

Published: 25 October 2019

Communicated by Kenneth S. Berenhaut

Authors

Loran Crowell
Eastern Kentucky University Richmond, KY United States

Steve Szabo
Eastern Kentucky University Richmond, KY United States

Vol. 12, No. 8, 2019