A053534 -id:A053534 - cp's OEIS frontend

A055545 Number of unlabeled matroids on n points.

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

Offset: 0

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

A058669 Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 36, 15, 1, 1, 31, 171, 171, 31, 1, 1, 63, 813, 2053, 813, 63, 1, 1, 127, 4012, 33442, 33442, 4012, 127, 1, 1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1, 1, 511

Offset: 0

N. J. A. Sloane, Dec 30 2000

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1;
  1,   1;
  1,   3,     1;
  1,   7,     7,       1;
  1,  15,    36,      15,       1;
  1,  31,   171,     171,      31,       1;
  1,  63,   813,    2053,     813,      63,     1;
  1, 127,  4012,   33442,   33442,    4012,   127,   1;
  1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 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

Row sums give A058673.
Columns include (truncated versions of) A000012 (k=0), A000225 (k=1), A058681 (k=2), A058687 (k=3).
Cf. A000079, A000110, A053534, A058710, A058711.

From Petros Hadjicostas, Oct 10 2019: (Start)
T(n,0) = 1 for n >= 0.
T(n,1) = 2^n - 1 for n >= 1. [Dukes (2004), Theorem 2.1 (ii).]
T(n,2) = Bell(n+1) - 2^n = A000110(n+1) - A000079(n) for n >= 2. [Dukes (2004), Theorem 2.1 (ii).]
T(n,k) = Sum_{m = k..n} binomial(n,m) * A058711(m,k) for n >= k. [Dukes (2004), see the equations before Theorem 2.1.]
(End)

A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.

Original entry on oeis.org

0, 0, 1, 3, 7, 13, 23, 37, 58, 87, 128, 183, 259, 359, 493, 668, 898, 1194, 1578, 2067, 2693, 3484, 4485, 5739, 7313, 9270, 11705, 14714, 18431, 22995, 28598, 35439, 43787, 53929, 66238, 81120, 99096, 120732, 146746, 177930, 215267, 259849, 313022, 376282

Offset: 0

N. J. A. Sloane, Dec 30 2000

nonn

Number of non-isomorphic rank-2 matroids over S_n.

Starting (1, 3, 7, 13, ...) = row sums of triangle A171239. - Gary W. Adamson, Dec 05 2009

Acketa, Dragan M. "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

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Muniru A Asiru)
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.
Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
Index entries for sequences related to matroids

Column k=2 of A053534.
Cf. A000041, A000065 (first differences), A000070.
Cf. A171239. - Gary W. Adamson, Dec 05 2009

List([1..41],n->Sum([1..n-1],k->NrPartitions(k)-1)); # Muniru A Asiru, Aug 10 2018

a:= proc(n) option remember; `if`(n<2, 0,
      combinat[numbpart](n)+a(n-1)-1)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 10 2019

With[{s = PartitionsP /@ Range[0, 40]}, MapIndexed[Total@ Take[s, First@ #2] - First@ #2 &, s]] (* Michael De Vlieger, Sep 04 2017 *)
With[{nn=50},#[[2]]-#[[1]]&/@Thread[{Range[0,nn],Accumulate[PartitionsP[Range[0,nn]]]}]]-1 (* Harvey P. Dale, Sep 05 2023 *)

G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Aug 10 2018

Name clarified by Ilya Gutkovskiy, Aug 10 2018

A256156 Triangular array of numbers of 2-polymatroids of rank k on n unlabeled points, for n>=0, 0<=k<=2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 10, 12, 10, 3, 1, 1, 4, 21, 49, 78, 49, 21, 4, 1, 1, 5, 39, 172, 584, 778, 584, 172, 39, 5, 1, 1, 6, 68, 573, 5236, 18033, 46661, 18033, 5236, 573, 68, 6, 1, 1, 7, 112, 1890, 72205, 971573, 149636721, 19498369, 149636721, 971573, 72205, 1890, 112, 7, 1

Offset: 0

Max Alekseyev, Mar 16 2015

The rows are symmetric: a(n,k) = a(n,2n-k).

Starting with n=7, the rows are not unimodal.

			Triangle starts with:
n=0: 1
n=1: 1 1 1
n=2: 1 2 4 2 1
n=3: 1 3 10 12 10 3 1
n=4: 1 4 21 49 78 49 21 4 1
n=5: 1 5 39 172 584 778 584 172 39 5 1
n=6: 1 6 68 573 5236 18033 46661 18033 5236 573 68 6 1
n=7: 1 7 112 1890 72205 971573 149636721 19498369 149636721 971573 72205 1890 112 7 1

Thomas J. Savitsky, Enumeration of 2-polymatroids on up to seven elements. SIAM J. Discrete Math., 28(4):1641-1650, 2014. arXiv:1401.8006

Cf. A256157, A256158, A256159, A053534

A058693 Number of nonisomorphic matroids of rank 3 on n labeled points.

Original entry on oeis.org

1, 4, 13, 38, 108, 325, 1275, 10037, 298491, 31899134

Offset: 3

N. J. A. Sloane, Dec 30 2000

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.
Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
Yoshitake Matsumoto, Database of Matroids
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
Index entries for sequences related to matroids

Column k=3 of A053534.

a(9) from Petros Hadjicostas, Oct 10 2019 using the papers of Mayhew and Royle
a(10)-a(11) from Manfred Scheucher, Sep 02 2020
a(12) from Database of Matroids added by Andrey Zabolotskiy, Mar 26 2023

A336704 Number of nonisomorphic matroids of rank 4 on n labeled points.

Original entry on oeis.org

1, 5, 23, 108, 940, 190214, 4886380924

Offset: 4

Manfred Scheucher, Aug 31 2020

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.
Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
Yoshitake Matsumoto, Database of Matroids

Column k=4 of A053534.

Cf. A058693.

a(10) from Database of Matroids added by Andrey Zabolotskiy, Mar 26 2023

cp's OEIS Frontend

A055545 Number of unlabeled matroids on n points.

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A058669 Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

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A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.

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A256156 Triangular array of numbers of 2-polymatroids of rank k on n unlabeled points, for n>=0, 0<=k<=2n.

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A058693 Number of nonisomorphic matroids of rank 3 on n labeled points.

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A336704 Number of nonisomorphic matroids of rank 4 on n labeled points.

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