Counting by 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 | References | Additional Information

Abstract: A by rectangular array is called a Latin rectangle if all the integers appear in each row of and if distinct integers occur in each column of . The number of by Latin rectangles is unknown for ; Riordan has given a formula for the number of by 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 by Latin rectangles by using Möbius inversion.

We include a table giving the approximate number of by Latin rectangles for . The table has exact values for .

**[1]**K. P. Bogart,*Closure relations and Möbius functions*(submitted).**[2]**H. Crapo,*Möbius inversion in lattices*, Arch. Math. (Basel)**19**(1968), 595-607 (1969). MR**39**#6791. MR**0245483 (39:6791)****[3]**C. Greene,*Combinatorial geometry*, Notes from the course given at N.S.F. Seminar in Combinatorics at Bowdoin College in 1971.**[4]**J. Riordan,*An introduction to combinatorial analysis*, Wiley Publications in Math. Statistics, Wiley, New York; Chapman & Hall, London, 1958. MR**20**#3077. MR**0096594 (20:3077)****[5]**G.-C. Rota,*On the foundations of combinatorial theory*. I,*Theory of Möbius functions*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**2**(1964), 340-368. MR**30**#4688. MR**0174487 (30:4688)**

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0389618-9

Article copyright:
© Copyright 1976
American Mathematical Society