A000474 -id:A000474 - 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

A350017 Number of isotopism classes containing symmetric unipotent reduced Latin squares of order 2n.

Original entry on oeis.org

1, 1, 1, 6, 396, 526915616, 1132835421602062347

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.

For odd n the terms equal A000474.

Cf. A350009.

A120488 Number of nonisomorphic perfect 1-factorizations of complete graph K_{2n}.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 23, 3155

Offset: 1

N. J. A. Sloane, Jul 22 2006

CRC Handbook Combin. Designs, p. 664.
Barbara M. Maenhaut, Perfect 1-factorizations of complete and complete bipartite graphs, talk given at 31st Australasian Conf. Combin. Math and Combin. Computing, Alice Springs, 2006.
Petrenyuk, L. and Petrenyuk, A.; Intersection of Perfect One-Factorizations of Complete Graphs, Cybernetics 16 (1980), 6-9.
Wallis, W. D.; 1-Factorizations of complete graphs, pp. 593-631 in J. H. Dinitz and D R. Stinson, Contemporary Design Theory, Wiley, 1992.

Jeffrey H. Dinitz and David K. Garnick, There are 23 Nonisomorphic Perfect One-Factorizations of K_{14}, J. Combin. Des. 4 (1996), no. 1, 1-3.
Wallis, W. D.; One-factorization of a graph, in Encyclopaedia of Mathematics (ed. Hazewinkel, Michiel), Springer-Verlag, 2002.

Cf. A000438, A000474, A120489.

Additional references from David Garnick (dgarnick(AT)gmail.com), Jan 17 2007
Edited by N. J. A. Sloane at the suggestion of Ian Wanless, Apr 01 2008
a(8) from Ian Wanless, Oct 20 2019

cp's OEIS Frontend

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

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

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A350017 Number of isotopism classes containing symmetric unipotent reduced Latin squares of order 2n.

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A120488 Number of nonisomorphic perfect 1-factorizations of complete graph K_{2n}.

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