A056642 -id:A056642 - cp's OEIS frontend

A001199 Erroneous version of A056642.

Original entry on oeis.org

1, 1, 2, 6, 32, 353, 8390, 436399, 50468754

Offset: 1

dead

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
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).

A001200 Number of linear geometries on n (unlabeled) points.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 24, 69, 384, 5250, 232929, 28872973

Offset: 0

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

For the labeled case see A056642.

Also a(n) = 1 + number of non-isomorphic simple rank-3 matroids on n elements (see A058731); a(n) = number of non-isomorphic 2-partitions of a set of size n. For 1-partitions see A000041.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #42.
CRC Handbook of Combinatorial Designs, 1996, pp. 216, 697.
J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points. Bull. Soc. Math. Belg. 19 1967 421-437.
P. Robillard, 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).

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.
A. Betten and D. Betten, Linear spaces with at most 12 points, J. Combinatorial Designs, Volume 7, 1999, pp. 119 - 145.
J. E. Blackburn, H. H. Crapo, and D. A. Higgs, A catalogue of combinatorial geometries, Math. Comp 27 1973 155-166.
J. Doyen, Sur le nombre d'espaces linéaires non isomorphes de n points [Annotated and scanned copy]
D. G. Glynn, Rings of geometries II, J. Combin. Theory, A 49 (1988), 26-66.
D. G. Glynn, A geometrical isomorphism algorithm, Bull. ICA 7 (1993), 36-38.
Robert Haas, Cographs, arXiv:1905.12627 [math.GM], 2019.
G. Heathcote, Linear spaces on 16 points, J. Combin. Designs, Vol. 1, No. 5 (1993), 359-378.
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).
Ch. Pietsch, On the classification of linear spaces of order 11, J. Comb. Designs, Vol. 3, No. 3 (1995), 185-193.

Cf. A000041, A056642, A058731, A001548.

A058710 Triangle T(n,k) giving number of loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 14, 11, 1, 0, 1, 51, 106, 26, 1, 0, 1, 202, 1232, 642, 57, 1, 0, 1, 876, 22172, 28367, 3592, 120, 1, 0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1

Offset: 0

N. J. A. Sloane, Dec 31 2000

From Petros Hadjicostas, Oct 10 2019: (Start)
The old references have some typos, some of which were corrected in the recent references (in 2004). Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51 (see the comment by Ralf Stephan below); T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.
This triangular array is the same as A058711 except that the current one has row n = 0 and column k = 0.
(End)

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1;
  0, 1;
  0, 1,    1;
  0, 1,    4,      1;
  0, 1,   14,     11,       1;
  0, 1,   51,    106,      26,      1;
  0, 1,  202,   1232,     642,     57,     1;
  0, 1,  876,  22172,   28367,   3592,   120,   1;
  0, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
  ...

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. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.

Cf. Same as A058711 (except for row n=0 and column k=0).
Row sums give A058712.
Columns include (truncated versions of) A000007 (k=0), A000012 (k=1), A058692 (k=2), A058715 (k=3).

From Petros Hadjicostas, Oct 10 2019: (Start)
T(n,0) = 0^n for n >= 0.
T(n,1) = 1 for n >= 1.
T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2.
T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.
T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]
(End)

T(5,2) corrected from 31 to 51 by Ralf Stephan, Nov 29 2004

A002773 Number of nonisomorphic simple matroids (or geometries) with n points.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 26, 101, 950, 376467

Offset: 0

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 138.
Knuth, Donald E. "The asymptotic number of geometries." Journal of Combinatorial Theory, Series A 16.3 (1974): 398-400.
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).

N. Bansal, R. Pendavingh, and J. G. van der Pol, On the number of matroids, arXiv:1206.6270v1 [math.CO], 2012.
Nikhil Bansal, Rudi A. Pendavingh, and Jorn G. van der Pol, On the number of matroids, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2013; full version in Combinatorica, 35:3 (2015), 253-277.
J. E. Blackburn, H. H. Crapo, and D. A. Higgs, A catalogue of combinatorial geometries, Math. Comp 27 (1973), 155-166.
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.
Henry H. Crapo and Gian-Carlo Rota, 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.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
M. J. Piff, An upper bound for the number of matroids, J. Combinatorial Theory Ser. B, vol 14 (1973), pp. 241-245.
Gordon Royle and Dillon Mayhew, 9-element matroids.
N. J. A. Sloane, Initial terms (* denotes 5 points in general position in 4-space).
Eric Weisstein's World of Mathematics, Matroid.
Index entries for sequences related to matroids

Cf. A055545, A056642. Row sums of A058730.

Limit_{ n -> oo }  (log_2 log_2 a(n))/n = 1. [Knuth]
2^n/n^(3/2) << log a(n) << 2^n/n, proved by Knuth and Piff respectively. - Charles R Greathouse IV, Mar 20 2021
Bansal, Pendavingh, & van der Pol prove an upper bound almost matching the lower bound above: log a(n) <= 2*sqrt(2/Pi)*2^n/n^(3/2)*(1 + o(1)). - Charles R Greathouse IV, Mar 20 2021

a(9) from Gordon Royle, Dec 23 2006

A058711 Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 14, 11, 1, 1, 51, 106, 26, 1, 1, 202, 1232, 642, 57, 1, 1, 876, 22172, 28367, 3592, 120, 1, 1, 4139, 803583, 8274374, 991829, 19903, 247, 1

Offset: 1

N. J. A. Sloane, Dec 31 2000

From Petros Hadjicostas, Oct 09 2019: (Start)
The old references had some typos, some of which were corrected in the recent ones. Few additional typos were corrected here from the recent references. Here are some of the changes: T(5,2) = 31 --> 51; T(5,4) = 21 --> 26; sum of row n=5 is 185 (not 160 or 165); T(8,3) = 686515 --> 803583; T(8, 6) = 19904 --> 19903, and some others.
This triangular array is the same as A058710 except that it has no row n = 0 and no column k = 0.
(End)

			Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,    1;
  1,    4,      1;
  1,   14,     11,       1;
  1,   51,    106,      26,      1;
  1,  202,   1232,     642,     57,     1;
  1,  876,  22172,   28367,   3592,   120,   1;
  1, 4139, 803583, 8274374, 991829, 19903, 247, 1;
  ...

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. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Index entries for sequences related to matroids

Same as A058710 (except for row n=0 and column k=0).
Row sums give A058712.
Columns include (truncated versions of) A000012 (k=1), A058692 (k=2), A058715 (k=3).

From Petros Hadjicostas, Oct 09 2019: (Start)
T(n,1) = 1 for n >= 1.
T(n,2) = Bell(n) - 1 = A000110(n) - 1 = A058692(n) for n >= 2.
T(n,3) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(i,3) for n >= 3.
T(n,k) = Sum_{i = k..n} Stirling2(n,i) * A058720(i,k) for n >= k. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.]
(End)

Several values corrected by Petros Hadjicostas, Oct 09 2019

A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 31, 16, 1, 1, 352, 337, 42, 1, 1, 8389, 18700, 2570, 99, 1, 1, 433038, 7642631, 907647, 16865, 219, 1

Offset: 2

N. J. A. Sloane, Dec 31 2000

			Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:
  1;
  1,      1;
  1,      5,       1;
  1,     31,      16,      1;
  1,    352,     337,     42,     1;
  1,   8389,   18700,   2570,    99,   1;
  1, 433038, 7642631, 907647, 16865, 219, 1;
  ...

Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
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. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g. [See p. 11.]
Index entries for sequences related to matroids

Cf. A000110 (Bell numbers), A002662, A058710, A058711, A058716, A058730, A100728.
Row sums give A058721.
Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).

From Petros Hadjicostas, Oct 09 2019: (Start)
T(n, n-1) = 2^n - 1 - binomial(n+1,2) = A002662(n) for n >= 2. [Dukes (2004), Lemma 2.2(i).]
T(n, n-2) = A100728(n) = A000110(n+1) + binomial(n+3,4) + 2*binomial(n+1,4) - 2^n - 2^(n-1)*binomial(n+1,2). [Dukes (2004), Lemma 2.2(iii).]
(End)

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

A058715 Number of loopless matroids of rank 3 on n labeled points.

Original entry on oeis.org

1, 11, 106, 1232, 22172, 803583, 70820187, 16122092568

Offset: 3

N. J. A. Sloane, Dec 31 2000

nonn

The sequence was updated based on more recent references by W. M. B. Dukes. The calculation of a(9) and a(10) depends on the values of A056642 for n = 9 and n = 10. Note that (A056642) - 1 is column k = 3 of A058720. - Petros Hadjicostas, Oct 09 2019

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. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Index entries for sequences related to matroids

Column k=3 of both A058710 and A058711 (which are the same except for column k=0).

Cf. A008277, A056442, A058720.

a(n) = Sum_{i = 3..n} Stirling2(n,i) * (A056642(i) - 1) = Sum_{i = 3..n} A008277(n,i) * A058720(n,3) for n >= 3. [Dukes (2004), p. 3; see the equation with the Stirling numbers of the second kind.] - Petros Hadjicostas, Oct 10 2019

a(8) corrected by and more terms from Petros Hadjicostas, Oct 09 2019

cp's OEIS Frontend

A001199 Erroneous version of A056642.

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A001200 Number of linear geometries on n (unlabeled) points.

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A058710 Triangle T(n,k) giving number of loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

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A002773 Number of nonisomorphic simple matroids (or geometries) with n points.

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A058711 Triangle T(n,k) giving the number of loopless matroids of rank k on n labeled points (n >= 1, 1 <= k <= n).

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A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

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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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A058715 Number of loopless matroids of rank 3 on n labeled points.

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