A000479 Number of 1-factorizations of K_{n,n}.

1, 1, 1, 2, 24, 1344, 1128960, 12198297600, 2697818265354240, 15224734061278915461120, 2750892211809148994633229926400, 19464657391668924966616671344752852992000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

Comments

Also, number of Latin squares of order n with first row 1,2,...,n.

Also number of fixed diagonal Latin squares of order n. - Eric W. Weisstein, Dec 18 2005

Also maximum number of Latin squares of order n such that no two of them have all the same rows (respectively, columns). - Rick L. Shepherd, Mar 01 2008

References

• CRC Handbook of Combinatorial Designs, 1996, p. 660.
• Denes and Keedwell, Latin Squares and Applications, Academic Press 1974.

Links

• B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
• B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
• Artur Schaefer, Endomorphisms of The Hamming Graph and Related Graphs, arXiv preprint arXiv:1602.02186 [math.CO], 2016.
• 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, 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)
• Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., Titov V.S., 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
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A000315, A000528, A002860.

See A040082 and A264603 for other versions.

Formula

a(n) = A000315(n)*(n-1)! = A002860(n)/n!.

Extensions

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