A058710 -id:A058710 - cp's OEIS frontend

A058692 a(n) = B(n) - 1, where B(n) = Bell numbers, A000110.

1, 4, 14, 51, 202, 876, 4139, 21146, 115974, 678569, 4213596, 27644436, 190899321, 1382958544, 10480142146, 82864869803, 682076806158, 5832742205056, 51724158235371, 474869816156750, 4506715738447322, 44152005855084345

Offset: 2

N. J. A. Sloane, Dec 30 2000

nonn

			G.f. = x^2 + 4*x^3 + 14*x^4 + 51*x^5 + 202*x^6 + 876*x^7 + 4139*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 2..200
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, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
Index entries for sequences related to matroids

Column k=2 of both A058710 and A058711 (which are the same except for column k=0).

Cf. A000110.

[Bell(n)-1: n in [2..30]]; // Vincenzo Librandi, Mar 04 2014

A058692 := proc(n)
    combinat[bell](n)-1 ;
end proc:
seq(A058692(n),n=2..40) ; # R. J. Mathar, May 25 2025

Table[BellB[n, 1] - 1, {n, 2, 23}] (* Zerinvary Lajos, Jul 16 2009 *)

G.f.: Sum_{k > 1} x^k / ((1 - x) * (1 - x^2) * ... * (1 - x^k)). - Michael Somos, Feb 26 2014

E.g.f.: exp(exp(x) - 1) - exp(x). - Ilya Gutkovskiy, Feb 08 2020

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)

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)

A058716 Triangle T(n,k) giving number of nonisomorphic 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, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 9, 4, 1, 0, 1, 10, 25, 18, 5, 1, 0, 1, 14, 70, 85, 31, 6, 1, 0, 1, 21, 217, 832, 288, 51, 7, 1

Offset: 0

N. J. A. Sloane, Dec 31 2000

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  4,   3,   1;
  0, 1,  6,   9,   4,   1;
  0, 1, 10,  25,  18,   5,  1;
  0, 1, 14,  70,  85,  31,  6, 1;
  0, 1, 21, 217, 832, 288, 51, 7, 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

Cf. A058717 (same except for border), A058710, A058711. Row sums give A058718. Diagonals give A000065, A058719.

Corrected and extended by Jean-François Alcover, Oct 21 2013

Reverted to original data by Sean A. Irvine, Aug 16 2022

A058712 Number of loopless matroids on n labeled points.

Original entry on oeis.org

1, 1, 2, 6, 27, 185, 2135, 55129, 10094077

Offset: 0

N. J. A. Sloane, Dec 31 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.
Index entries for sequences related to matroids

Row sums of A058710.

a(5) corrected by Petros Hadjicostas, Oct 09 2019

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

A058717 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 1, 1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 10, 25, 18, 5, 1, 1, 14, 70, 85, 31, 6, 1, 1, 21, 217, 832, 288, 51, 7, 1

Offset: 1

N. J. A. Sloane, Dec 31 2000

			1;
1,  1;
1,  2,   1;
1,  4,   3,   1;
1,  6,   9,   4,   1;
1, 10,  25,  18,   5,  1;
1, 14,  70,  85,  31,  6, 1;
1, 21, 217, 832, 288, 51, 7, 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.
Index entries for sequences related to matroids
W. M. B. Dukes, On the number of matroids on a finite set

Cf. A058716 (same except for border), A058710, A058711.

Row sums give A058718. Diagonals give A000065, A058719.

Corrected and extended by Jean-François Alcover, Oct 21 2013

Reverted to original data by Jean-François Alcover, Aug 17 2022

cp's OEIS Frontend

A058692 a(n) = B(n) - 1, where B(n) = Bell numbers, A000110.

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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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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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A058716 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

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A058712 Number of loopless matroids on n labeled points.

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

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A058717 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 1, 1<=k<=n).

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