A002773 -id:A002773 - 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

A055545 Number of unlabeled matroids on n points.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 98, 306, 1724, 383172

Offset: 0

Eric W. Weisstein

This is the total number of pairwise non-isomorphic (i.e., "unlabeled") matroids on n points, with no restrictions on loops, parallel elements etc.

Partial sums of A058718. - Jonathan Vos Post, Apr 25 2010

J. G. Oxley, Matroid Theory. Oxford, England: Oxford University Press, 1993. See p. 473.

Dragan M. Acketa, On the enumeration of matroids of rank-2, Zbornik radova Prirodnomatematickog fakulteta-Univerzitet u Novom Sadu 8 (1978): 83-90. - _N. J. A. Sloane_, Dec 04 2022
Jayant Apte and J. M. Walsh, Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding, arXiv preprint arXiv:1605.04598 [cs.IT], 2016-2017.
Jesus DeLoera, Yvonne Kemper, and Steven Klee, h-vectors of small matroid complexes, arXiv:1106.2576 [math.CO], 2011.
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.
S. C. Locke, Matroids
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.
Gordon Royle and Dillon Mayhew, 9-element matroids.
Eric Weisstein's World of Mathematics, Matroid.
Eric Weisstein's World of Mathematics, Graph Vertex.
D. J. A. Welsh, A bound for the number of matroids, J. Combinat. Theory, Ser. A, 6 (1969), 313-316. - From _N. J. A. Sloane_, May 06 2012
Index entries for sequences related to matroids

Cf. A002773, A058673 (labeled matroids), A058718.

Row sums of A053534.

a(9) from Gordon Royle, Dec 23 2006

Name clarified by Lorenzo Sauras Altuzarra, Aug 10 2023

A058730 Triangle T(n,k) giving number of nonisomorphic simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 11, 4, 1, 1, 23, 49, 22, 5, 1, 1, 68, 617, 217, 40, 6, 1, 1, 383, 185981, 188936, 1092, 66, 7, 1, 1, 5249, 4884573865

Offset: 2

N. J. A. Sloane, Dec 31 2000

To make this sequence a triangular array, we assume n >= 2 and 2 <= k <= n. According to the references, however, we have T(0,0) = T(1, 1) = 1, and 0 in all other cases. - Petros Hadjicostas, Oct 09 2019

			Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:
  1;
  1,   1;
  1,   2,      1;
  1,   4,      3,      1;
  1,   9,     11,      4,    1;
  1,  23,     49,     22,    5,  1;
  1,  68,    617,    217,   40,  6, 1;
  1, 383, 185981, 188936, 1092, 66, 7, 1;
  ...
From _Petros Hadjicostas_, Oct 09 2019: (Start)
Matsumoto et al. (2012, p. 36) gave an incomplete row n = 10 (starting at k = 2):
  1, 5249, 4884573865, *, 4886374072, 9742, 104, 8, 1;
They also gave incomplete rows for n = 11 and n = 12.
(End)

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]
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.
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. [See Table 2, p. 9.]
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431. [See Table 2, p. 420.]
Y. Matsumoto, S. Moriyama, H. Imai, and D. Bremmer, Matroid enumeration for incidence geometry, Discrete Comput. Geom. 47 (2012), 17-43.
Gordon Royle and Dillon Mayhew, 9-element matroids.
Index entries for sequences related to matroids

Cf. A058720. Row sums give A002773.

Columns include (truncations of) A000012 (k=2), A058731 (k=3), A058733 (k=4).

From Petros Hadjicostas, Oct 09 2019: (Start)
T(n, n-1) = n-2 for n >= 2. [Dukes (2004), Lemma 2.2(ii).]
T(n, n-2) = 6 - 4*n + Sum_{k = 1..n} A000041(k) for n >= 3. [Dukes (2004), Lemma 2.2(iv).]
(End)

Row n=9 from Petros Hadjicostas, Oct 09 2019 using the papers by Mayhew and Royle

A114572 Number of "ultrasweet" Boolean functions of n variables which depend on all the variables.

Original entry on oeis.org

2, 1, 2, 6, 27, 185, 2135, 55129

Offset: 0

Don Knuth, Aug 17 2008, Oct 14 2008

nonn

Inverse binomial transform of A114491.
This sequence enumerates a certain type of matroid, except for the first entry (which is 2 instead of 1). If the first entry is changed from 2 to 1, giving A118085, this enumerates "combinatorial geometries" on n labeled points.
These are matroids in which no element has rank 0; equivalently, all one-element sets are independent; equivalently, the closure of the empty set is empty.
These are called "simple matroids" in A002773. So A118085 is the "labeled" equivalent of that sequence, which counts unlabeled points.

			For all n>1, a function like "x2" is counted in A114491 but not in the present sequence.

Cf. A114302, A114303, A114491, A118085, A002773.

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

A058733 Number of nonisomorphic simple matroids of rank 4 on n labeled points.

Original entry on oeis.org

1, 3, 11, 49, 617, 185981

Offset: 4

N. J. A. Sloane, Dec 31 2000

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]
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.
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. [See Table 2, p. 9.]
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431. [See Table 2, p. 420.]
Gordon Royle and Dillon Mayhew, 9-element matroids.
Index entries for sequences related to matroids

Column k=4 of A058730.

a(9) from Petros Hadjicostas, Oct 09 2019 using the papers by Mayhew and Royle

cp's OEIS Frontend

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

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A055545 Number of unlabeled matroids on n points.

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A058730 Triangle T(n,k) giving number of nonisomorphic simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

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A114572 Number of "ultrasweet" Boolean functions of n variables which depend on all the variables.

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

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A058733 Number of nonisomorphic simple matroids of rank 4 on n labeled points.

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