A040082 Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n.

1, 1, 1, 2, 2, 22, 564, 1676267, 115618721533, 208904371354363006, 12216177315369229261482540

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

Comments

Here "isotopy class" means an equivalence class of Latin squares under the operations of row permutation, column permutation and symbol permutation. [Brendan McKay]

References

• R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
• 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).

Links

• J. W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184. [a(7) and a(8) appear to be given incorrectly. - _N. J. A. Sloane_, Jan 23 2020]
• A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
• G. Kolesova, C. W. H. Lam and L. Thiel, On the number of 8x8 Latin squares, J. Combin. Theory,(A) 54 (1990) 143-148.
• Brendan D. McKay, Latin Squares (has list of all such squares)
• Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
• Brendan D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
• Eduard Vatutin, Alexey Belyshev, Stepan Kochemazov, Oleg Zaikin, Natalia Nikitina, Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing, Russian Supercomputing Days (Суперкомпьютерные дни в России), 2018.
• Eric Weisstein's World of Mathematics, Latin Square
• M. B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

Cf. A002860, A003090, A000315. See A000528 for another version.

Extensions

7 X 7 and 8 X 8 results confirmed by Brendan McKay

Beware: erroneous versions of this sequence can be found in the literature!

a(9)-a(10) (from the McKay-Meynert-Myrvold article) from Richard Bean, Feb 17 2004

a(11) from Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009