A001200 -id:A001200 - cp's OEIS frontend

A056642 Number of linear spaces on n (labeled) points.

1, 1, 2, 6, 32, 353, 8390, 433039, 50166354, 13480967630

Offset: 1

W. M. B. Dukes (dukes(AT)stp.dias.ie), Aug 28 2000

Alternatively, number of linear geometries on n (labeled) points. For the unlabeled case see A001200.

Also a(n) = 1 + number of simple rank-3 matroids on n (labeled) elements; a(n) = number of 2-partitions of a set of size n.

L. M. Batten and A. Beutelspacher: The theory of finite linear spaces, Cambridge Univ. Press, 1993 (see the Appendix).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points, Bull. Soc. Math. Belg. 19 (1967), 421-437.
J. A. Thas, Sur le nombre d'espaces linéaires non isomorphes de n points, Bull. Soc. Math. Belg. 21 (1969), 57-66.

W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. Dukes, Bounds on the number of generalized partitions and some applications, Australas. J. Combin. 28 (2003), 257-261.
W. M. B. Dukes, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
Index entries for sequences related to matroids

Corrected version of A001199. Cf. A002773, A001200, A031436, A058731.

a(9) and a(10) from Gordon Royle, May 29 2006

A001548 Number of connected linear spaces with n (unlabeled) points.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 4, 13, 42, 308, 4845, 227613, 28639650

Offset: 0

N. J. A. Sloane

Euler transform is A001200. - Michael Somos, Apr 24 2014

In any linear space any two distinct points belong to exactly one line. A linear space is disconnected if there exists a partition of the points of the space into two subsets such that for any two distinct points in a subset of the partition the unique line they both belong to is completely contained in that subset. - Michael Somos, Apr 24 2014

			a(2) = 0 because the unique linear space on two points can be partitioned into two single point subsets which disconnects the space vacuously. a(5) = 2 because there are two connected linear spaces with 5 points: one has only one line and the other has two lines with three points that intersect in one point that belongs to no other line while the other four points belong to three lines. - _Michael Somos_, Apr 24 2014

L. M. Batten and A. Beutelspacher: The theory of finite linear spaces, Cambridge Univ. Press, 1993 (see the Appendix).
Doyen, Jean; Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Doyen, Sur le nombre d'espaces lineaires non isomorphes de n points [Annotated and scanned copy]
Olaf Lechtenfeld, Konrad Schwerdtfeger and Johannes Thürigen, N=4 Multi-Particle Mechanics, WDVV Equation and Roots, SIGMA 7 (2011), 023. See the table on p. 18 for the sequence a(n)-1 for n > 2.
Pierre Robillard, On the weighted finite linear spaces, Bull. Soc. Math. Belg. 22 (1970), 227-241. [Annotated and scanned copy]

Cf. A001200, A056642.

A001200 = Cases[Import["https://oeis.org/A001200/b001200.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[A001200 // Rest] (* Jean-François Alcover, Jan 04 2020, updated Mar 17 2020 *)

More terms could be obtained from A056642. - N. J. A. Sloane, Jul 26 2004
a(10)-a(12) from A001200. - Michael Somos, Apr 24 2014
a(12) corrected by Jean-François Alcover, Jan 04 2020

A031436 Number of proper linear spaces of order n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 3, 7, 1, 119, 398, 161925, 2412890

Offset: 0

Anton Betten (Anton.Betten(AT)uni-bayreuth.de)

Anton Betten and Dieter Betten, The proper linear spaces on 17 points, Discrete Applied Mathematics, Volume 95, no. 1-3, 1999, pp. 83-108.
Anton Betten and Dieter Betten, Note on the Proper Linear Spaces on 18 Points, in "Algebraic Combinatorics and Applications", Springer 2001, pp. 40-54.
Anton Betten and Dieter Betten, Data up to 18 points, Proceedings of ALCOMA 1999, Springer Verlag 2000, 40-54 [free access].
Hans-Dietrich O. F. Gronau, Ronald C. Mullin, Christian Pietsch, Pairwise Balanced Designs as Linear Spaces, pp. 228-235, table 4.19 [but beware errors], in: Charles J. Colbourn, Ed., CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton 1996 [PDF preview]

Cf. A001200, A031437, A031438, A001439, A056642.

Additional comments from Michael Somos, Nov 18 2001

A058731 Number of nonisomorphic simple matroids of rank 3 on n unlabeled points.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 9, 23, 68, 383, 5249, 232928, 28872972

Offset: 0

N. J. A. Sloane, Dec 31 2000; May 28 2006

Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
Crapo, Henry H.; Rota, Gian-Carlo; On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 1970 109-133. [Annotated scanned copy of pages 126 and 127 only]
W. M. B. Dukes, Tables of matroids
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
Index entries for sequences related to matroids

Equals A001200 - 1 (see that entry for further information).

A diagonal of A058730.

Definition corrected by Gordon Royle, Feb 13 2007

A031437 Number of nonisomorphic regular linear spaces RLIN(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 3, 4, 9, 14, 33, 839, 2041, 22192

Offset: 0

Anton Betten (Anton.Betten(AT)uni-bayreuth.de)

A. Betten and D. Betten: Regular Linear Spaces, Beitraege Algebra Geometrie 38 (1): 111-124, 1997.
A. Betten and D. Betten: The proper linear spaces on 17 points, Discrete Applied Mathematics, Volume 95, no. 1-3, 1999, pp. 83-108.
CRC Handbook of Combinatorial Designs, in the article by Gronau, Mullin and Pietsch.

Cf. A001200, A031436.

A002876 Number of weighted linear spaces of total weight n.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 85, 239

Offset: 1

N. J. A. Sloane

Robillard, Pierre; On the weighted finite linear spaces. Bull. Soc. Math. Belg. 22 (1970), 227-241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robillard, Pierre, On the weighted finite linear spaces, Bull. Soc. Math. Belg. 22 (1970), 227-241. [Annotated and scanned copy]

Cf. A001200, A002877.

A002877 Number of connected weighted linear spaces of total weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 35, 116

Offset: 1

N. J. A. Sloane

Robillard, Pierre; On the weighted finite linear spaces. Bull. Soc. Math. Belg. 22 (1970), 227-241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robillard, Pierre, On the weighted finite linear spaces, Bull. Soc. Math. Belg. 22 (1970), 227-241. [Annotated and scanned copy]

Cf. A001200, A002876.

A031438 Number of nonisomorphic proper regular linear spaces, PRLIN(n).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 3, 3, 0, 84, 25, 0

Offset: 0

Anton Betten (Anton.Betten(AT)uni-bayreuth.de)

A. Betten and D. Betten: The proper linear spaces on 17 points, Discrete Applied Mathematics, Volume 95, no. 1-3, 1999, pp. 83-108.

Cf. A001200, A031436, A031437.

A309114 Minimal linear spaces on n points.

Original entry on oeis.org

1, 0, 1, 3, 8, 23, 208

Offset: 3

Michael De Vlieger, Jul 13 2019

Batten, Lynn Margaret, and Beutelspacher, Albrecht. The Theory of Finite Linear Spaces: Combinatorics of Points and Lines, Cambridge University Press, 1993.

Robert Haas, Cographs, arXiv:1905.12627 [math.GM], 2019, p. 46.

Cf. A001200, A001403, A001548, A309115.

A309115 Minimal linear spaces with n nontrivial lines.

Original entry on oeis.org

1, 2, 5, 16

Offset: 1

Michael De Vlieger, Jul 13 2019

Per Haas, a(5) >= 70.

Batten, Lynn Margaret, and Beutelspacher, Albrecht. The Theory of Finite Linear Spaces: Combinatorics of Points and Lines, Cambridge University Press, 1993.

Robert Haas, Cographs, arXiv:1905.12627 [math.GM], 2019, p. 46.

Cf. A001200, A001403, A001548, A309114.

cp's OEIS Frontend

A056642 Number of linear spaces on n (labeled) points.

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A001548 Number of connected linear spaces with n (unlabeled) points.

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A031436 Number of proper linear spaces of order n.

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A058731 Number of nonisomorphic simple matroids of rank 3 on n unlabeled points.

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A031437 Number of nonisomorphic regular linear spaces RLIN(n).

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A002876 Number of weighted linear spaces of total weight n.

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A002877 Number of connected weighted linear spaces of total weight n.

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A031438 Number of nonisomorphic proper regular linear spaces, PRLIN(n).

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A309114 Minimal linear spaces on n points.

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A309115 Minimal linear spaces with n nontrivial lines.

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