A000315 Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.

1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more

Comments

A reduced Latin square of order n is an n X n matrix where each row and column is a permutation of 1..n and the first row and column are 1..n in increasing order. - Michael Somos, Mar 12 2011

The Stones-Wanless (2010) paper shows among other things that a(n) is 0 mod n if n is composite and 1 mod n if n is prime.

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
• J. Denes, A. D. Keedwell, editors, Latin Squares: new developments in the theory and applications, Elsevier, 1991, pp. 1, 388.
• R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.
• C. R. Rao, S. K. Mitra and A. Matthai, editors, Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India, 1966, p. 193.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
• H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, pp. 37, 53.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.

Links

• S. E. Bammel and J. Rothstein, The number of 9x9 Latin squares, Discrete Math., 11 (1975), 93-95.
• Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.
• Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Quad Squares, arXiv:2308.07455 [math.HO], 2023.
• B. Cherowitzo, Latin Squares, Comb. Structures Lecture Notes.
• Gheorghe Coserea, Solutions for n=5.
• Gheorghe Coserea, Solutions for n=6.
• Gheorghe Coserea, MiniZinc model for generating solutions.
• E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014
• Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
• B. D. McKay, A. Meynert and W. Myrvold, Small latin squares, quasigroups and loops, J. Combin. Designs, vol. 15, no. 2 (2007) pp. 98-119.
• B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
• B. D. McKay and I. M. Wanless, On the number of Latin squares. Preprint 2004.
• B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
• Young-Sik Moon, Jong-Yoon Yoon, Jong-Seon No, and Sang-Hyo Kim, Interference Alignment Schemes Using Latin Square for Kx3 MIMO X Channel, arXiv:1810.05400 [cs.IT], 2018.
• Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021. Mentions this sequence.
• J. Shao and W. Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
• D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
• E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157. (in Russian)
• E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares. CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
• Eric Weisstein's World of Mathematics, Latin Square.
• M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
• Index entries for sequences related to Latin squares and rectangles
• Index entries for sequences related to quasigroups

Crossrefs

Cf. A000479, A002860, A003090, A040082, A057771, A057997.

Formula

a(n) = A002860(n) / (n! * (n-1)!) = A000479(n) / (n-1)!.

Extensions

Added June 1995: the 10th term was probably first computed by Eric Rogoyski

a(11) (from the McKay-Wanless article) from Richard Bean, Feb 17 2004