A023994 -id:A023994 - cp's OEIS frontend

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

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

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.

			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.

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.

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)

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

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.

A098702 Number of self-polar configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 10, 25, 95, 365, 1432, 5799, 24092, 102413, 445363, 1991320

Offset: 1

N. J. A. Sloane, Nov 05 2004

			Example: the Fano plane is the only 7_3 configuration and it is self-polar.

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, Discrete Comput. Geom., 35 (2006), 405-427.
T. Pisanski, M. Boben, D. Marušic, A. Orbanic, A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265-276.

Cf. A001403, A023994.

a(1)-a(18) from the Betten, Brinkmann and Pisanski article.

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

A098822 Number of cyclic configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 2, 3, 4, 4, 2, 5, 3

Offset: 1

N. J. A. Sloane, Nov 05 2004

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.
T. Pisanski, M. Boben, D. Marušic, A. Orbanic, A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265-276.

Cf. A001403, A023994.

a(1)-a(18) from the Betten, Brinkmann and Pisanski article.

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

A098804 Number of point-transitive configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 4, 2, 3, 5, 6, 2, 9

Offset: 1

N. J. A. Sloane, Nov 05 2004

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.

Cf. A001403, A023994.

A098841 Number of disconnected (decomposable) configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 13, 47

Offset: 1

N. J. A. Sloane, Nov 05 2004

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.

Cf. A001403, A023994.

A098852 Number of triangle-free self-dual configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 6, 40, 307

Offset: 1

N. J. A. Sloane, Nov 05 2004

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.

Cf. A001403, A100001, A023994.

A098854 Number of triangle-free self-polar configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 6, 40, 303

Offset: 1

N. J. A. Sloane, Nov 05 2004

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.

Cf. A001403, A098702, A023994.

A099999 Number of geometrical configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 31, 229

Offset: 1

N. J. A. Sloane, following correspondence from Branko Grünbaum and Tomaz Pisanski, Nov 12 2004

A geometrical configuration of type (n_3) consists of a set of n points in the Euclidean or extended Euclidean plane together with a set of n lines, such that each point belongs to 3 lines and each line contains 3 points.

Branko Grünbaum comments that it would be nice to settle the question as to whether all combinatorial configurations (13_3) are (as he hopes) geometrically realizable.

			The smallest examples occur for n = 9, where there are three configurations, one of which is the configuration arising from Pappus's Theorem (see the World of Mathematics "Configuration" link for drawings of all three).
The configuration arising from Desargues's theorem (see link above to an illustration) is one of the nine configurations for n = 10.

Many of the following references refer to combinatorial configurations (A001403) rather than geometrical configurations, but are included here in case they are helpful.
A. Betten and D. Betten, Regular linear spaces, Beitraege zur Algebra und Geometrie, 38 (1997), 111-124.
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.

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.
H. Gropp, Configurations and their realization, Discr. Math. 174 (1997), 137-151.
Jim Loy, Desargues's Theorem
Jim Loy, The configuration (10_3) arising from Desargues's theorem
Tomo Pisanski, Papers on configurations
T. Pisanski, M. Boben, D. Marušic, A. Orbanic and A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265-276.
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.

Cf. A001403 (abstract or combinatorial configurations (n_3)), A023994, A100001, A098702, A098804, A098822, A098841, A098851, A098852, A098854.

A100001 Number of self-dual combinatorial configurations of type (n_3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 10, 25, 95, 366, 1433, 5802, 24105, 102479, 445577, 1992044

Offset: 1

N. J. A. Sloane, Nov 05 2004

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.

Interchanging the roles of points and lines gives the dual configuration. A configuration is self-dual if there is an isomorphism from it to its dual.

			Example: the Fano plane is the only (7_3) configuration and is self-dual. It contains 7 points 1,2,...7 and 7 triples, 124, 235, 346, 457, 561, 672, 713.
The unique (8_3) configuration is also self-dual. It consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v_3, Discrete Appl. Math., 99 (2000), 331-338.
T. Pisanski, M. Boben, D. Marušic, A. Orbanic, A. Graovac, The 10-cages and derived configurations, Discrete Math. 275 (2004), 265-276.

Cf. A001403 (configurations (n_3), with many further references), A099999, A023994.

a(1)-a(18) from the Betten, Brinkmann and Pisanski article.

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

cp's OEIS Frontend

A001403 Number of combinatorial configurations of type (n_3).

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A098702 Number of self-polar configurations of type (n_3).

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A098822 Number of cyclic configurations of type (n_3).

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A098804 Number of point-transitive configurations of type (n_3).

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A098841 Number of disconnected (decomposable) configurations of type (n_3).

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A098852 Number of triangle-free self-dual configurations of type (n_3).

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A098854 Number of triangle-free self-polar configurations of type (n_3).

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A099999 Number of geometrical configurations of type (n_3).

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A100001 Number of self-dual combinatorial configurations of type (n_3).

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