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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Counting $ 3$ by $ n$ Latin rectangles


Authors: K. P. Bogart and J. Q. Longyear
Journal: Proc. Amer. Math. Soc. 54 (1976), 463-467
DOI: https://doi.org/10.1090/S0002-9939-1976-0389618-9
MathSciNet review: 0389618
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Abstract: A $ k$ by $ n$ rectangular array $ A$ is called a Latin rectangle if all the integers $ 1,2, \ldots ,n$ appear in each row of $ A$ and if $ k$ distinct integers occur in each column of $ A$. The number of $ k$ by $ n$ Latin rectangles is unknown for $ k \geqslant 4$; Riordan has given a formula for the number of $ 3$ by $ n$ rectangles in terms of the solutions of the derangement (or displacement) problem and the menage problem. In this paper we derive an elementary formula for the number of $ 3$ by $ n$ Latin rectangles by using Möbius inversion.

We include a table giving the approximate number of $ 3$ by $ n$ Latin rectangles for $ n \leqslant 20$. The table has exact values for $ n \leqslant 11$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0389618-9
Article copyright: © Copyright 1976 American Mathematical Society