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

1,9

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.

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.

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

Table of n, a(n) for n=1..19.

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)

Example

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.

Crossrefs

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

Sequence in context: A079522 A325579 A034016 * A072136 A286444 A080406

Adjacent sequences: A001400 A001401 A001402 * A001404 A001405 A001406

Keyword

nonn,nice,hard,more

Author

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

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.

Status

approved