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A001403 Number of combinatorial configurations of type (n_3). 12
0, 0, 0, 0, 0, 0, 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, 3004881, 38904499, 530452205, 7640941062 (list; graph; refs; listen; history; text; internal format)



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.


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.


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, 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.

Jim Loy, Mathematics Page (see Desargues's Theorem)

Jim Loy, The configuration (10_3) arising from Desargues's theorem

Tomo Pisanski, Papers on configurations

B. Sturmfels and N. White, All 11_3 and 12_3 configurations are rational, Aeq. Math., 39 1990 254-260.

Von Sterneck, Die Config. 11_3, Monat. f. Math. Phys., 5 325-330 1894.

Von Sterneck, Die Config. 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 is the familiar configuration arising from Desargues's theorem (see Loy illustration).


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




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


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.



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Last modified May 15 07:17 EDT 2021. Contains 343909 sequences. (Running on oeis4.)