A000528 -id:A000528

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A040082

Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n.
(Formerly M0392 N0150)

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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

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

CROSSREFS

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

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane

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

STATUS

approved

A000479

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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

Table of n, a(n) for n=0..11.

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

FORMULA

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

CROSSREFS

Cf. A000315, A000528, A002860.

See A040082 and A264603 for other versions.