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%I #31 Mar 03 2023 07:59:11
%S 1,0,1,0,0,0,1,0,1,0,0,0,2,0,80,0,0,0,11084874829,0,14796207517873771
%N Number of nonisomorphic Steiner triple systems (STS's) S(2,3,n) on n points.
%C a(n) also counts the following objects:
%C isomorphism classes of idempotent totally symmetric Latin squares of order n,
%C isotopism classes containing idempotent totally symmetric Latin squares of order n,
%C species containing idempotent totally symmetric Latin squares of order n,
%C isomorphism classes of totally symmetric loops of order n+1,
%C isomorphism classes of totally symmetric unipotent Latin squares of order n+1,
%C isomorphism classes containing totally symmetric reduced Latin squares of order n+1,
%C isotopism classes containing totally symmetric unipotent Latin squares of order n+1,
%C isotopism classes containing totally symmetric reduced Latin squares of order n+1,
%C species containing totally symmetric unipotent Latin squares of order n+1, and
%C species containing totally symmetric reduced Latin squares of order n+1.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 304.
%D CRC Handbook of Combinatorial Designs, 1996, p. 70.
%H Daniel Heinlein and Patric R. J. Östergård, <a href="https://arxiv.org/abs/2303.01207">Enumerating Steiner Triple Systems</a>, arXiv:2303.01207 [math.CO], 2023.
%H Petteri Kaski and Patric R. J. Östergård (Patric.Ostergard(AT)hut.fi), <a href="http://www.tcs.hut.fi/~pkaski/sts19.ps">The Steiner triple systems of order 19</a>.
%H Petteri Kaski and Patric R. J. Östergård, <a href="https://doi.org/10.1090/S0025-5718-04-01626-6">The Steiner triple systems of order 19</a>, Mathematics of Computation, Vol. 73, No. 248 (Oct., 2004), pp. 2075-2092.
%H Brendan D. McKay and Ian M. Wanless, <a href="https://doi.org/10.1002/jcd.21814">Enumeration of Latin squares with conjugate symmetry</a>, J. Combin. Des. 30 (2022), 105-130.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SteinerTripleSystem.html">Steiner Triple System</a>.
%H <a href="/index/St#Steiner">Index entries for sequences related to Steiner systems</a>.
%Y Cf. A001201, A030128, A051390, A124118, A124119, A076019.
%K nonn,nice,hard,more
%O 1,13
%A _Eric W. Weisstein_
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