A035481 -id:A035481 - cp's OEIS frontend

A000438 Number of 1-factorizations of complete graph K_{2n}.

Original entry on oeis.org

1, 1, 6, 6240, 1225566720, 252282619805368320, 98758655816833727741338583040

Offset: 1

N. J. A. Sloane

CRC Handbook of Combinatorial Designs (see pages 655, 720-723).
N. T. Gridgeman, Latin Squares Under Restriction and a Jumboization, J. Rec. Math., 5 (1972), 198-202.
W. D. Wallis, 1-Factorizations of complete graphs, pp. 593-631 in Jeffrey H. Dinitz and D. R. Stinson, Contemporary Design Theory, Wiley, 1992.

Jeffrey H. Dinitz, David K. Garnick, and Brendan D. McKay, There are 526,915,620 nonisomorphic one-factorizations of K_{12}, J. Combin. Des. 2 (1994), no. 4, 273-285.
Alan Hartman, and Alexander Rosa, Cyclic one-factorization of the complete graph, European J. Combin. 6 (1985), no. 1, 45-48.
Dieter Jungnickel, and Vladimir D. Tonchev, Counting Steiner triple systems with classical parameters and prescribed rank, arXiv:1709.06044 [math.CO], 2017.
Petteri Kaski, and Patric R. J. Östergård, There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of K14, Journal of Combinatorial Designs 17 (2009) 147-159.
Mario Krenn, Xuemei Gu, and Anton Zeilinger, Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings, arXiv:1705.06646 [quant-ph], 2017 and Phys. Rev. Lett. 119, 240403, 2017. [Mario Krenn said in an email, "We would not have discovered this connection between quantum mechanical experiments and graph theory, thus the physical interpretations and all the generalisations we are developing right now, without you and A000438."]
D. V. Zinoviev, On the number of 1-factorizations of a complete graph [in Russian], Problemy Peredachi Informatsii, 50 (No. 4), 2014, 71-78.
Index entries for sequences related to tournaments

Cf. A000474, A003191, A035481, A035483. Equals A036981 / (2n+1)!.

For K_16 the answer is approximately 1.48 * 10^44 and for K_18 1.52 * 10^63. - Dinitz et al.

a(7) found by Patric Östergård and Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 19 2007

A036981 Number of (2n+1) X (2n+1) symmetric matrices each of whose rows is a permutation of 1..(2n+1).

Original entry on oeis.org

1, 6, 720, 31449600, 444733651353600, 10070314878246926155776000, 614972203951464612786852376432607232000

Offset: 0

Joshua Zucker and Joe Keane

Number of different schedules for 2n+2 teams. - Andres Cardemil (andrescarde(AT)yahoo.com), Nov 28 2001

Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018-2021.
Index entries for sequences related to tournaments.

Cf. A000438, A000474, A035481, A036980, A064120, A065594.

a(n) = A000438(n+1) * (2*n+1)!.

a(5)-a(6) computed from A000438 by Max Alekseyev, Jun 17 2011

A057992 Number of commutative quasigroups of order n.

Original entry on oeis.org

1, 1, 1, 3, 7, 11, 491, 6381, 10940111, 1225586965, 130025302505741, 252282619993126717, 2209617218725712597768722, 98758655816833782283724345637

Offset: 0

Christian G. Bower, Nov 01 2000

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.
Index entries for sequences related to quasigroups

Cf. A057991-A057994, A057771, A057996, A035481, A350009, A350010.

Added a(7) = 6381, W. Edwin Clark, Jan 04 2011

a(8)-a(13) from Ian Wanless, Dec 08 2021

A350009 a(n) is the number of isotopism classes containing symmetric Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 7, 415, 3460, 35878418, 6320290037, 4612965997149292, 15859695832489637513

Offset: 1

Ian Wanless, Dec 08 2021

Isotopism classes are obtained by permuting rows, permuting columns and permuting symbols. There is a stronger notion of equivalence called "species" (also known as main classes and paratopism classes). For this particular problem the counts for species equal the counts for isotopism classes.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

The odd terms agree with A350010. Cf. A035481

A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

Original entry on oeis.org

1, 1, 4, 456, 10936320, 130025295912960, 2209617218725251404267520

Offset: 0

Joshua Zucker and Joe Keane

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(2*n). - Max Alekseyev, Apr 23 2010

a(5)-a(6) from Ian Wanless, Oct 20 2019

A089925 Number of commutative loops (quasigroups with an identity element) of order n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 8, 17, 2265, 30583, 358335026, 69522550106, 55355570223093935, 206176045800229002160

Offset: 0

Christian G. Bower and Juergen Buntrock (jubu(AT)jubu.com), Nov 14 2003

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.
Index entries for sequences related to quasigroups

Cf. A057771, A035481, A350009, A350010, A057992.

a(9) from Michael Thwaites (michael.thwaites(AT)ucop.edu), Jan 23 2004
a(0) prepended by Jianing Song, Oct 26 2019
a(10)-a(13) from Ian Wanless, Dec 08 2021

A350010 a(n) is the number of equivalence classes of symmetric Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 7, 423, 3460, 35878510, 6320290037, 4612966007179768, 15859695832489637513

Offset: 1

Ian Wanless, Dec 08 2021

Equivalence here includes permutation of the symbols as well as simultaneously applying one permutation to both the rows and columns.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

The odd terms agree with A350009. Cf. A035481.

A036980 Number of (2n) X (2n) symmetric matrices each of whose rows is a permutation of 1..(2n).

Original entry on oeis.org

1, 2, 96, 328320, 440952422400

Offset: 0

Joshua Zucker and Joe Keane

Cf. A035481, A036981.

a(n) = A035483(n) * (2n)!

A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

Original entry on oeis.org

1, 1, 2, 6, 96, 720, 328320, 31449600, 440952422400, 444733651353600, 471835793808949248000, 10070314878246926155776000, 1058410183156945383046388908032000, 614972203951464612786852376432607232000

Offset: 0

Joshua Zucker and Joe Keane

The even and odd subsequences are A036980, A036981.

			a(3) = 6 because the first row is arbitrary (say, 213) and the rest is then determined. By symmetry the second row has to be 132 or 123 but in order for the third row/column to work it has to be 132.

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(n) * n!. [From Max Alekseyev, Apr 23 2010]

a(10)-a(13) (using A035481) from Alois P. Heinz, May 05 2023

A111341 Number of Latin squares in dimension n with first row and first column 1,2,3,...,n that are non-associative but commutative (element ij = element ji).

Original entry on oeis.org

0, 0, 0, 0, 0, 396, 6120, 10934400, 1225559160, 130025295822240, 252282619805005440, 2209617218725251390961920, 98758655816833727741298666240

Offset: 1

Artur Jasinski, Nov 06 2005

At order 6 there are commutative but non-associative Latin squares. - Artur Jasinski, Dec 08 2006

a(n) = A035481(n) - A058162(n).

a(5) corrected and a(8)-a(13) added by Max Alekseyev, Jul 23 2025

cp's OEIS Frontend

A000438 Number of 1-factorizations of complete graph K_{2n}.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A036981 Number of (2n+1) X (2n+1) symmetric matrices each of whose rows is a permutation of 1..(2n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A057992 Number of commutative quasigroups of order n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A350009 a(n) is the number of isotopism classes containing symmetric Latin squares of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

Views

Author

Keywords

Crossrefs

Formula

Extensions

A089925 Number of commutative loops (quasigroups with an identity element) of order n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A350010 a(n) is the number of equivalence classes of symmetric Latin squares of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

A036980 Number of (2n) X (2n) symmetric matrices each of whose rows is a permutation of 1..(2n).

Views

Author

Keywords

Crossrefs

Formula

A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A111341 Number of Latin squares in dimension n with first row and first column 1,2,3,...,n that are non-associative but commutative (element ij = element ji).

Views

Author

Keywords

Comments

Formula

Extensions