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%I #50 Jun 28 2023 16:01:44
%S 1,1,6,6240,1225566720,252282619805368320,
%T 98758655816833727741338583040
%N Number of 1-factorizations of complete graph K_{2n}.
%D CRC Handbook of Combinatorial Designs (see pages 655, 720-723).
%D N. T. Gridgeman, Latin Squares Under Restriction and a Jumboization, J. Rec. Math., 5 (1972), 198-202.
%D W. D. Wallis, 1-Factorizations of complete graphs, pp. 593-631 in Jeffrey H. Dinitz and D. R. Stinson, Contemporary Design Theory, Wiley, 1992.
%H Jeffrey H. Dinitz, David K. Garnick, and Brendan D. McKay, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4085">There are 526,915,620 nonisomorphic one-factorizations of K_{12}</a>, J. Combin. Des. 2 (1994), no. 4, 273-285.
%H Alan Hartman, and Alexander Rosa, <a href="https://doi.org/10.1016/S0195-6698(85)80020-2">Cyclic one-factorization of the complete graph</a>, European J. Combin. 6 (1985), no. 1, 45-48.
%H Dieter Jungnickel, and Vladimir D. Tonchev, <a href="https://arxiv.org/abs/1709.06044">Counting Steiner triple systems with classical parameters and prescribed rank</a>, arXiv:1709.06044 [math.CO], 2017.
%H Petteri Kaski, and Patric R. J. Östergård, <a href="https://doi.org/10.1002/jcd.20188">There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of K14</a>, Journal of Combinatorial Designs 17 (2009) 147-159.
%H Mario Krenn, Xuemei Gu, and Anton Zeilinger, <a href="http://arxiv.org/abs/1705.06646">Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings</a>, arXiv:1705.06646 [quant-ph], 2017 and <a href="https://doi.org/10.1103/PhysRevLett.119.240403">Phys. Rev. Lett. 119, 240403</a>, 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."]
%H D. V. Zinoviev, <a href="https://www.mathnet.ru/eng/ppi2154">On the number of 1-factorizations of a complete graph</a> [in Russian], Problemy Peredachi Informatsii, 50 (No. 4), 2014, 71-78.
%H <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%Y Cf. A000474, A003191, A035481, A035483. Equals A036981 / (2n+1)!.
%K nonn,hard,more,nice
%O 1,3
%A _N. J. A. Sloane_
%E For K_16 the answer is approximately 1.48 * 10^44 and for K_18 1.52 * 10^63. - Dinitz et al.
%E a(7) found by Patric Östergård and Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 19 2007
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