A000479 -id:A000479

Search: a000479 -id:a000479

Displaying 1-10 of 10 results found.

page 1

A094050

Duplicate of A000479.

+20
0

1, 1, 2, 24, 1344, 1128960

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..6.

CROSSREFS

Cf. A000479.

KEYWORD

dead

STATUS

approved

A002860

Number of Latin squares of order n; or labeled quasigroups.
(Formerly M2051 N0812)

+10
39

1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000

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

OFFSET

1,2

COMMENTS

Also the number of minimum vertex colorings in the n X n rook graph. - Eric W. Weisstein, Mar 02 2024

REFERENCES

David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.

Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.

H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.

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.

Ronald Alter, Research Problems: How Many Latin Squares are There?, Amer. Math. Monthly 82 (1975), no. 6, 632-634. MR1537769.

S. E. Bammel and J. Rothstein, The number of 9 X 9 Latin squares, Discrete Math., 11 (1975), 93-95.

D. Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248. See p. 3241.

Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.

Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.

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

Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Quad Squares, arXiv:2308.07455 [math.HO], 2023.

Hai-Dang Dau and Nicolas Chopin, Waste-free Sequential Monte Carlo, arXiv:2011.02328 [stat.CO], 2020.

Abdelrahman Desoky, Hany Ammar, Gamal Fahmy, Shaker El-Sappagh, Abdeltawab Hendawi, and Sameh H. Basha, Latin Square and Artificial Intelligence Cryptography for Blockchain and Centralized Systems, Int'l Conf. Adv. Intel. Sys. Informat., Proc. 9th Int'l Conf. (AISI 2023) pp. 444-455.

Thangavelu Geetha, Amritanshu Prasad, and Shraddha Srivastava, Schur Algebras for the Alternating Group and Koszul Duality, arXiv:1902.02465 [math.RT], 2019.

E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence. - N. J. A. Sloane, Mar 15 2014

Yue Guan, Minjia Shi and Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.

Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.

Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.

A.-A. A. Jucys, The number of distinct Latin squares as a group-theoretical constant, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 265-272. MR0419259 (54 #7283).

Dieter Jungnickel and Vladimir D. Tonchev, Counting Steiner triple systems with classical parameters and prescribed rank, arXiv:1709.06044 [math.CO], 2017.

Lintao Liu, Xuehu Yan, Yuliang Lu, and Huaixi Wang, 2-threshold Ideal Secret Sharing Schemes Can Be Uniquely Modeled by Latin Squares, National University of Defense Technology, Hefei, China, (2019).

B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Des. 15 (2007), no. 2, 98-119.

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

B. D. McKay and I. M. Wanless, On the number of Latin squares. Preprint 2004.

B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.

J. Shao and W. Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.

Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018.

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.

Eric Weisstein's World of Mathematics, Latin Square.

Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.

Eric Weisstein's World of Mathematics, Rook Graph.

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

Krasimir Yordzhev, The bitwise operations in relation to obtaining Latin squares, arXiv preprint arXiv:1605.07171 [cs.OH], 2016.

Index entries for sequences related to Latin squares and rectangles

Index entries for sequences related to quasigroups

FORMULA

a(n) = n!*A000479(n) = n!*(n-1)!*A000315(n).

MATHEMATICA

Table[Length[ResourceFunction["FindProperColorings"][GraphProduct[CompleteGraph[n], CompleteGraph[n], "Cartesian"], n]], {n, 5}]

CROSSREFS

Cf. A000315, A000479.

Cf. A003090, A040082, A057991.

Cf. A098679 (Latin cubes).

A row of the array in A249026.

KEYWORD

hard,nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

One more term (from the McKay-Wanless article) from Richard Bean, Feb 17 2004

STATUS

approved

A000315

Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.
(Formerly M3690 N1508)

+10
24

1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840

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

OFFSET

1,4

COMMENTS

A reduced Latin square of order n is an n X n matrix where each row and column is a permutation of 1..n and the first row and column are 1..n in increasing order. - Michael Somos, Mar 12 2011

The Stones-Wanless (2010) paper shows among other things that a(n) is 0 mod n if n is composite and 1 mod n if n is prime.

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.

J. Denes, A. D. Keedwell, editors, Latin Squares: new developments in the theory and applications, Elsevier, 1991, pp. 1, 388.

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

C. R. Rao, S. K. Mitra and A. Matthai, editors, Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India, 1966, p. 193.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.

H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, pp. 37, 53.

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, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.

LINKS

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

S. E. Bammel and J. Rothstein, The number of 9x9 Latin squares, Discrete Math., 11 (1975), 93-95.

Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.

B. Cherowitzo, Latin Squares, Comb. Structures Lecture Notes.

Gheorghe Coserea, Solutions for n=5.

Gheorghe Coserea, Solutions for n=6.

Gheorghe Coserea, MiniZinc model for generating solutions.

E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence. - N. J. A. Sloane, Mar 15 2014

Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.

B. D. McKay, A. Meynert and W. Myrvold, Small latin squares, quasigroups and loops, J. Combin. Designs, vol. 15, no. 2 (2007) pp. 98-119.

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

B. D. McKay and I. M. Wanless, On the number of Latin squares. Preprint 2004.

B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.

Young-Sik Moon, Jong-Yoon Yoon, Jong-Seon No, and Sang-Hyo Kim, Interference Alignment Schemes Using Latin Square for Kx3 MIMO X Channel, arXiv:1810.05400 [cs.IT], 2018.

Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021. Mentions this sequence.

J. Shao and W. Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

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, and 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)

E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, 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.

M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]

Index entries for sequences related to Latin squares and rectangles

Index entries for sequences related to quasigroups

FORMULA

a(n) = A002860(n) / (n! * (n-1)!) = A000479(n) / (n-1)!.

CROSSREFS

Cf. A000479, A002860, A003090, A040082, A057771, A057997.

KEYWORD

nonn,hard,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

Added June 1995: the 10th term was probably first computed by Eric Rogoyski

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

STATUS

approved

A274171

Number of diagonal Latin squares of order n with the first row in order.

+10
13

1, 0, 0, 2, 8, 128, 171200, 7447587840, 5056994653507584

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

OFFSET

1,4

COMMENTS

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Sep 29 2020

LINKS

Table of n, a(n) for n=1..9.

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.

M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project

Eduard I. Vatutin, a(9) value fixed after

E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)

E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)

E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).

E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.

Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S.Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.

E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.

Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).

Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and 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.

E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)

Index entries for sequences related to Latin squares and rectangles

FORMULA

a(n) = A274806(n)/n!.

EXAMPLE

The a(4) = 2 diagonal Latin squares are:

0 1 2 3 0 1 2 3

2 3 0 1 3 2 1 0

3 2 1 0 1 0 3 2

1 0 3 2 2 3 0 1

The a(5) = 8 diagonal Latin squares are:

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

1 3 4 2 0 1 4 3 0 2 2 3 4 0 1 2 4 1 0 3

4 2 1 0 3 3 2 1 4 0 4 0 1 2 3 4 0 3 2 1

2 0 3 4 1 4 3 0 2 1 1 2 3 4 0 3 2 4 1 0

3 4 0 1 2 2 0 4 1 3 3 4 0 1 2 1 3 0 4 2

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

3 4 0 1 2 3 4 1 2 0 4 2 0 1 3 4 2 3 0 1

1 2 3 4 0 4 2 3 0 1 1 4 3 2 0 3 4 1 2 0

4 0 1 2 3 2 0 4 1 3 3 0 1 4 2 1 3 0 4 2

2 3 4 0 1 1 3 0 4 2 2 3 4 0 1 2 0 4 1 3

CROSSREFS

Cf. A000315, A000479, A274806, A287764, A309283.

KEYWORD

nonn,more,hard

AUTHOR

Eduard I. Vatutin, Jul 07 2016

EXTENSIONS

a(9) added from Vatutin et al. (2016) by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

STATUS

approved

A274806

Number of diagonal Latin squares of order n.

+10
11

1, 0, 0, 48, 960, 92160, 862848000, 300286741708800, 1835082219864832081920

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

OFFSET

1,4

COMMENTS

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Oct 05 2020

LINKS

Table of n, a(n) for n=1..9.

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.

M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project

E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian).

Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.

Eduard I. Vatutin, a(9) value fixed

E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, 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.

Index entries for sequences related to Latin squares and rectangles

FORMULA

a(n) = A274171(n) * n!.

CROSSREFS

Cf. A000315, A000479, A274171, A287764, A309283.

KEYWORD

nonn,more,hard

AUTHOR

Eduard I. Vatutin, Jul 07 2016

EXTENSIONS

a(9) from Vatutin et al. (2016) added by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

STATUS

approved

A000528

Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.

+10
2

1, 1, 1, 2, 2, 17, 324, 842227, 57810418543, 104452188344901572, 6108088657705958932053657

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

OFFSET

1,4

COMMENTS

Here "type" means an equivalence class of Latin squares under the operations of row permutation, column permutation, symbol permutation and transpose. In the 1-factorizations formulation, these operations are labeling of left side, labeling of right side, permuting the order in which the factors are listed and swapping the left and right sides, respectively. - Brendan McKay

There are 6108088657705958932053657 isomorphism classes of one-factorizations of K_{11,11}. - Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009

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=1..11.

A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219

B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs, to appear (2005).

Index entries for sequences related to Latin squares and rectangles

CROSSREFS

See A040082 for another version.

Cf. A002860, A003090, A000315, A040082, A000479.

KEYWORD

hard,nonn,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Richard Bean, Feb 17 2004

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

STATUS

approved

A114631

Number of even fixed diagonal Latin squares.

+10
2

1, 1, 0, 24, 384, 702720, 6231859200, 1364560466411520

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

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..8.

Eric Weisstein's World of Mathematics, Alon-Tarsi Paradox

CROSSREFS

Cf. A000479, A114632.

KEYWORD

nonn,hard

AUTHOR

Eric W. Weisstein, Dec 18 2005

STATUS

approved

A114632

Number of odd fixed diagonal Latin squares.

+10
2

0, 0, 2, 0, 960, 426240, 5966438400, 1333257798942720

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..8.

Eric Weisstein's World of Mathematics, Alon-Tarsi Paradox

CROSSREFS

Cf. A000479, A114631.

KEYWORD

nonn,hard

AUTHOR

Eric W. Weisstein, Dec 18 2005

STATUS

approved

A344664

a(n) is the number of preference profiles in the stable marriage problem with n men and n women where both the men's and the women's preferences form a Latin square when arranged in a matrix. In addition, it is possible to arrange all people into n man-woman couples such that they rank each other first.

+10
2

1, 2, 24, 13824, 216760320, 917676490752000, 749944260264355430400000, 293457967200879687743551498616832000, 84112872283641495670736269523436185936222748672000, 27460610008848610956892895086773773421767179663217968124264448000000

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

OFFSET

1,2

COMMENTS

Two people who rank each other first are called soulmates. Thus, the profiles in this sequence have n pairs of soulmates.

The profiles with n pairs of soulmates are counted by sequence A343698. The profiles such that the men's preferences form a Latin square are counted by A343696. The profiles such that both men's and women's preferences form a Latin square are counted by A343697. The profiles in this sequence are the intersection of profiles in A343698 and A343697.

Both the men- and the women-proposing Gale-Shapley algorithm on the preference profiles described by this sequence end in one round.

LINKS

Table of n, a(n) for n=1..10.

Wikipedia, Gale-Shapley algorithm.

FORMULA

a(n) = A002860(n)^2 / n!.

a(n) = A000479(n) * A002860(n).

EXAMPLE

For n = 3, there are A002860(3) = 12 Latin squares of order 3. Thus, there are A002860(3) = 12 ways to set up the men's preference profiles. After that, the women's preference profiles form a Latin square with a fixed first column, as the first column is uniquely defined to generate 3 pairs of soulmates. Thus, there are A002860(3)/3! = 12/6 = 2 ways to set up the women's preference profiles, making a(3) = 12 * 2 = 24 preference profiles.

CROSSREFS

Cf. A000479, A002860, A185141, A343696, A343697, A343698, A344665.

KEYWORD

nonn

AUTHOR

Tanya Khovanova and MIT PRIMES STEP Senior group, Jun 01 2021

EXTENSIONS

Corrected by Tanya Khovanova, Aug 17 2021

STATUS

approved

A264603

Number of structurally distinct Latin squares of order n.

+10
1

1, 1, 1, 12, 192, 145164, 1524901344

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

OFFSET

1,4

COMMENTS

"Structurally distinct" means that the squares cannot be made identical by means of rotation, reflection, and/or permutation of the symbols. For other notions of distinctness, see A000479 and A040082.

REFERENCES

Hendrik Willem Barink, Email to N. J. A. Sloane, Nov 22 2015

LINKS

Table of n, a(n) for n=1..7.

Hendrik Willem Barink, How many structurally different latin squares do exist?

Gabriel Crudele, Peter Dukes, and Jonathan A. Noel, Six Permutation Patterns Force Quasirandomness, arXiv:2303.04776 [math.CO], 2023.

Mathematics Stack Exchange, How many structurally different latin squares of order 5 exist?

Mathematics Stack Exchange, How many structurally distinct Latin squares are there of order 7?

Index entries for sequences related to Latin squares and rectangles

CROSSREFS

Cf. A000479, A040082.

KEYWORD