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A000438 Number of 1-factorizations of complete graph K_{2n}. 9
1, 1, 6, 6240, 1225566720, 252282619805368320, 98758655816833727741338583040 (list; graph; refs; listen; history; text; internal format)



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.

D. V. Zinoviev, On the number of 1-factorizations of a complete graph [in Russian], Problemy Peredachi Informatsii, 50 (No. 4), 2014, 71-78.


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

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

Petteri Kaski, 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, 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."]

Index entries for sequences related to tournaments


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

Sequence in context: A202969 A003191 A298272 * A061109 A321983 A219014

Adjacent sequences: A000435 A000436 A000437 * A000439 A000440 A000441




N. J. A. Sloane


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



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