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A040082 Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n.
(Formerly M0392 N0150)
1, 1, 1, 2, 2, 22, 564, 1676267, 115618721533, 208904371354363006, 12216177315369229261482540 (list; graph; refs; listen; history; text; internal format)



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


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).


Table of n, a(n) for n=1..11.

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


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

Sequence in context: A212847 A087405 A001012 * A014358 A093355 A122962

Adjacent sequences:  A040079 A040080 A040081 * A040083 A040084 A040085




N. J. A. Sloane


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



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Last modified October 28 04:06 EDT 2020. Contains 338048 sequences. (Running on oeis4.)