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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]


J. W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184.

R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.

A. Hulpke, P. Kaski and P. R. J. Ostergard, The number of Latin squares of order 11, Preprint, 2009.

G. Kolesova, C. W. H. Lam and L. Thiel, On the number of 8x8 Latin squares, J. Combin. Theory,(A) 54 (1990) 143-148.

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).

M. B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.


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

B. D. McKay, Latin Squares (has list of all such squares)

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 E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Eric Weisstein's World of Mathematics, Latin Square

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!

Two more terms (from the McKay-Meynert-Myrvold article) from Richard Bean (rwb(AT)eskimo.com), Feb 17 2004

There are 12216177315369229261482540 isotopy classes of Latin squares of order 11. - Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009



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Last modified August 16 17:17 EDT 2017. Contains 290626 sequences.