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%I #55 Oct 30 2023 01:54:36
%S 0,0,0,0,0,0,1,1,3,10,31,229,2036,21399,245342,3004881,38904499,
%T 530452205,7640941062
%N Number of combinatorial configurations of type (n_3).
%C A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
%D Bokowski and Sturmfels, Comput. Synthetic Geom., Lect Notes Math. 1355, p. 41.
%D CRC Handbook of Combinatorial Designs, 1996, p. 255.
%D Branko Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, 103 (2009), American Mathematical Society.
%D D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, NY, 1952, Ch. 3.
%D F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
%D Pisanski, T. and Randic, M., Bridges between Geometry and Graph Theory, in Geometry at Work: Papers in Applied Geometry (Ed. C. A. Gorini), M.A.A., Washington, DC, pp. 174-194, 2000.
%D B. Polster, A Geometrical Picture Book, Springer, 1998, p. 28.
%D Sturmfels and White, Rational realizations..., in H. Crapo et al. editors, Symbolic Computation in Geometry, IMA preprint, Univ. Minn., 1988.
%D David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, p. 72.
%H A. Betten and D. Betten, <a href="http://www.math.colostate.edu/~betten/pub/betten5.pdf">Regular linear spaces</a>, Beiträge zur Algebra und Geometrie, 38 (1997), 111-124.
%H A. Betten and D. Betten, <a href="https://doi.org/10.1007/BF01225670">Tactical decompositions and some configurations v_4</a>, J. Geom. 66 (1999), 27-41.
%H A. Betten, G. Brinkmann and T. Pisanski, <a href="https://doi.org/10.1016/S0166-218X(99)00143-2">Counting symmetric configurations v_3</a>, Discrete Appl. Math., 99 (2000), 331-338.
%H M. Boben et al., <a href="https://web.archive.org/web/20220331033535/http://preprinti.imfm.si/PDF/00938.pdf">Small triangle-free configurations of points and lines</a>, Preprint series, Vol. 42 (2004), 938, University of Ljubljana.
%H M. Boben et al., <a href="https://doi.org/10.1007/s00454-005-1224-9">Small triangle-free configurations of points and lines</a>, Discrete Comput. Geom., 35 (2006), 405-427.
%H Jürgen Bokowski and Vincent Pilaud, <a href="https://arxiv.org/abs/1210.0306">Enumerating topological (n_k)-configurations</a>, arXiv:1210.0306 [cs.CG], 2012.
%H H. Gropp, <a href="http://dx.doi.org/10.1016/S0012-365X(96)00327-5">Configurations and their realization</a>, Discr. Math. 174 (1997), 137-151.
%H Nathan Kaplan, Susie Kimport, Rachel Lawrence, Luke Peilen and Max Weinreich, <a href="https://doi.org/10.1007/s00022-017-0391-1">Counting arcs in projective planes via Glynn’s algorithm.</a> J. Geom. 108, No. 3, 1013-1029 (2017).
%H Jim Loy, <a href="https://web.archive.org/web/20140108030301/http://www.jimloy.com:80/geometry/desargue.htm">Desargues's Theorem</a>
%H Jim Loy, <a href="/A099999/a099999.gif">The configuration (10_3) arising from Desargues's theorem</a>
%H Tomo Pisanski, <a href="https://web.archive.org/web/20050320084801/http://www.ijp.si/Configurations2004/papers.html">Presentations from International workshop Configurations 2004</a>
%H B. Sturmfels and N. White, <a href="https://doi.org/10.1007/BF01833153">All 11_3 and 12_3 configurations are rational</a>, Aeq. Math., 39 1990 254-260.
%H Robert Daublebsky von Sterneck, <a href="https://doi.org/10.1007/BF01691614">Die Configurationen 11_3</a>, Monat. f. Math. Phys., 5 325-330 1894.
%H Robert Daublebsky von Sterneck, <a href="https://doi.org/10.1007/BF01696586">Die Configurationen 12_3</a>, Monat. f. Math. Phys., 6 223-255 1895.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Configuration.html">Configuration.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Configuration_(geometry)">Configuration (geometry)</a>
%e The Fano plane is the only (7_3) configuration. It contains 7 points 1,2,...,7 and 7 triples, 124, 235, 346, 457, 561, 672, 713.
%e The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
%e There are three configurations (9_3), one of which arises from Pappus's theorem. See the World of Mathematics "Configuration" link above for diagrams of all three.
%e There are nine configurations (10_3), one of which is the familiar configuration arising from Desargues's theorem (see Loy illustration), which are realizable by straight lines on the plane, plus one non-realizable configuration - see Gropp's fig. 4 for a drawing of that configuration with almost straight lines.
%Y Cf. A023994, A099999 (geometrical configurations), A100001 (self-dual configurations), A098702, A098804, A098822, A098841, A098851, A098852, A098854.
%K nonn,nice,hard,more
%O 1,9
%A _N. J. A. Sloane_, D.Glynn(AT)math.canterbury.ac.nz
%E Von Sterneck has 228 instead of 229. His error was corrected by Gropp. The n=15 term was computed by Dieter and Anton Betten, University of Kiel.
%E a(16)-a(18) from the Betten, Brinkmann and Pisanski article.
%E a(19) from the Pisanski et al. article.
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