Divisors of the number of Latin rectangles

Journal of Combinatorial Theory. Series A

Volumn 117, Issue 2, 2010, Pages 204-215

  Stones, Douglas S ^a   Wanless, Ian M ^a  

^a MONASH UNIVERSITY   (Australia)

Author keywords

Latin rectangles; Latin squares

Indexed keywords

EID: 76449095025     PISSN: 00973165     EISSN: 10960899     Source Type: Journal    
DOI: 10.1016/j.jcta.2009.03.019     Document Type: Article

References (44)

1. Alter R. How many Latin squares are there?. Amer. Math. Monthly 82 (1975) 632-634
2. r, s). European J. Combin. 1 (1980) 9-17
3. Bailey R.A. Latin squares with highly transitive automorphism groups. J. Aust. Math. Soc. 33 (1982) 18-22
4. Bogart K.P., and Longyear J.Q. Counting 3 by n Latin rectangles. Proc. Amer. Math. Soc. 54 (1976) 463-467
5. Carlitz L. Congruences connected with three-line Latin rectangles. Proc. Amer. Math. Soc. 4 1 (1953) 9-11
6. Denés J., and Mullen G.L. Enumeration formulas for Latin and frequency squares. Discrete Math. 111 (1993) 157-163
7. Doyle P.G. The number of Latin rectangles (2007). arXiv:math/0703896v1 [math.CO] 15 pp
8. Drisko A.A. On the number of even and odd Latin squares of order p + 1. Adv. Math. 128 1 (1997) 20-35
9. r p. Electron. J. Combin. 5 1 (1998) R28 5 pp
10. Dulmage A.L. E650. Amer. Math. Monthly 52 8 (1945) 458
11. Dulmage A.L., and McMaster G.E. A formula for counting three-line Latin rectangles. Congr. Numer. 14 (1975) 279-289
12. Euler L. Recherches sur une nouvelle espéce de quarrés magiques. Verh. Zeeuwsch. Gennot. Weten. Vliss. 9 (1782) 85-239 Eneström E530, Opera Omnia OI7, 291-392
13. Euler L. Solutio quaestionis curiosae ex doctrina combinationum. Mémoires de l'Académie des Sciences de St. Pétersbourg 3 (1811) 57-64 E738, OI7, 435-440
14. Fu Z.-L. The number of Latin rectangles. Math. Practice Theory (2) (1992) 40-41 (in Chinese)
15. Gessel I.M. Counting three-line Latin rectangles. In: Labelle G., and Leroux P. (Eds). Proc. of the Colloque de Combinatoire Énumérative (1986), Springer
16. Gessel I.M. Counting Latin rectangles. Bull. Amer. Math. Soc. 16 1 (1987) 79-83
17. Godsil C.D., and McKay B.D. Asymptotic enumeration of Latin rectangles. J. Combin. Theory Ser. B 48 (1990) 19-44
18. Goulden I.P., and Jackson D.M. Combinatorial Enumeration (1983), Wiley
19. Hamilton J.R., and Mullen G.L. How many i - j reduced Latin rectangles are there?. Amer. Math. Monthly 87 5 (1980) 392-394
20. Jacob S.M. The enumeration of the Latin rectangle of depth three by means of a formula of reduction, with other theorems relating to non-clashing substitutions and Latin squares. Proc. London Math. Soc. (2) 31 (1930) 329-354
21. Jucys A.-A.A. The number of distinct Latin squares as a group-theoretical constant. J. Combin. Theory Ser. A 20 3 (1976) 265-272
22. Kerawala S.M. The enumeration of the Latin rectangle of depth three by means of a difference equation. Bull. Calcutta Math. Soc. 33 (1941) 119-127
23. Kerawala S.M. The asymptotic number of three-deep Latin rectangles. Bull. Calcutta Math. Soc. 39 (1947) 71-72
24. Kuznetsov N.Y. Estimating the number of Latin rectangles by the fast simulation method. Cybernet. Systems Anal. 45 1 (2009) 69-75
25. Light Jr. F.W. A procedure for the enumeration of 4 × n Latin rectangles. Fibonacci Quart. 11 3 (1973) 241-246
26. Light Jr. F.W. Enumeration of truncated Latin rectangles. Fibonacci Quart. 17 1 (1979) 34-36
27. MacMahon P.A. A new method in combinatory analysis, with applications to Latin squares and associated questions. Trans. Camb. Philos. Soc. 16 (1898) 262-290
28. MacMahon P.A. Combinatory Analysis (1960), Chelsea
29. McKay B.D., Meynert A., and Myrvold W. Small Latin squares, quasigroups, and loops. J. Combin. Des. 15 (2007) 98-119
30. McKay B.D., and Rogoyski E. Latin squares of order ten. Electron. J. Combin. 2 (1995) N3, 4 pp
31. McKay B.D., and Wanless I.M. On the number of Latin squares. Ann. Comb. 9 (2005) 335-344
32. Mullen G.L. How many i - j reduced Latin squares are there?. Amer. Math. Monthly 82 (1978) 751-752
33. J.R. Nechvatal, Counting Latin rectangles, PhD thesis, University of Southern California, 1979
34. Nechvatal J.R. Asymptotic enumeration of generalized Latin rectangles. Util. Math. 20 (1981) 273-292
35. Pranesachar C.R. Enumeration of Latin rectangles via SDR's. In: Dold A., Eckmann B., and Rao S.B. (Eds). Combinatorics and Graph Theory (1981), Springer
36. Riordan J. Three-line Latin rectangles. Amer. Math. Monthly 51 8 (1944) 450-452
37. Riordan J. Three-line Latin rectangles-II. Amer. Math. Monthly 53 1 (1946) 18-20
38. Riordan J. A recurrence relation for three-line Latin rectangles. Amer. Math. Monthly 59 3 (1952) 159-162
39. Shao J.-Y., and Wei W.-D. A formula for the number of Latin squares. Discrete Math. 110 (1992) 293-296
40. Sloane N.J.A. The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/
41. van Leijenhorst D.C. Symmetric functions, Latin squares and Van der Corput's "scriptum 3". Expo. Math. 18 5 (2000) 343-356
42. Yamamoto K. Asymptotic number of Latin rectangles and the symbolic method. Sûgaku 2 (1949)
43. Yamamoto K. An asymptotic series for the number of three-line Latin rectangles. J. Math. Soc. Japan 1 2 (1950) 226-241
44. Yamamoto K. Symbolic methods in the problem of three-line Latin rectangles. J. Math. Soc. Japan 5 1 (1953) 13-23