A001403 Number of combinatorial configurations of type (n_3).

0, 0, 0, 0, 0, 0, 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, 3004881, 38904499, 530452205, 7640941062

Offset: 1

Author

N. J. A. Sloane, D.Glynn(AT)math.canterbury.ac.nz

Keywords

• nonn
• nice
• hard
• more

Comments

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.

Examples

			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.
The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
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.
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.
		

References

• Bokowski and Sturmfels, Comput. Synthetic Geom., Lect Notes Math. 1355, p. 41.
• CRC Handbook of Combinatorial Designs, 1996, p. 255.
• Branko Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, 103 (2009), American Mathematical Society. See Table 2.2.1, page 69.
• D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, NY, 1952, Ch. 3.
• F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
• 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.
• B. Polster, A Geometrical Picture Book, Springer, 1998, p. 28.
• Sturmfels and White, Rational realizations..., in H. Crapo et al. editors, Symbolic Computation in Geometry, IMA preprint, Univ. Minn., 1988.
• David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, p. 72.

Links

• A. Betten and D. Betten, Regular linear spaces, Beiträge zur Algebra und Geometrie, 38 (1997), 111-124.
• A. Betten and D. Betten, Tactical decompositions and some configurations v_4, J. Geom. 66 (1999), 27-41.
• A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.
• M. Boben et al., Small triangle-free configurations of points and lines, Preprint series, Vol. 42 (2004), 938, University of Ljubljana.
• M. Boben et al., Small triangle-free configurations of points and lines, Discrete Comput. Geom., 35 (2006), 405-427.
• Jürgen Bokowski and Vincent Pilaud, Enumerating topological (n_k)-configurations, arXiv:1210.0306 [cs.CG], 2012.
• H. Gropp, Configurations and their realization, Discr. Math. 174 (1997), 137-151.
• Nathan Kaplan, Susie Kimport, Rachel Lawrence, Luke Peilen and Max Weinreich, Counting arcs in projective planes via Glynn's algorithm, J. Geom. 108, No. 3, 1013-1029 (2017).
• Jim Loy, Desargues's Theorem
• Jim Loy, The configuration (10_3) arising from Desargues's theorem
• Tomo Pisanski, Presentations from International workshop Configurations 2004
• B. Sturmfels and N. White, All 11_3 and 12_3 configurations are rational, Aeq. Math., 39 1990 254-260.
• Robert Daublebsky von Sterneck, Die Configurationen 11_3, Monat. f. Math. Phys., 5 325-330 1894.
• Robert Daublebsky von Sterneck, Die Configurationen 12_3, Monat. f. Math. Phys., 6 223-255 1895.
• Eric Weisstein's World of Mathematics, Configuration.
• Wikipedia, Configuration (geometry)

Crossrefs

Cf. A023994, A099999 (geometrical configurations), A100001 (self-dual configurations), A098702, A098804, A098822, A098841, A098851, A098852, A098854.

Extensions

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.

a(16)-a(18) from the Betten, Brinkmann and Pisanski article.

a(19) from the Pisanski et al. article.