A058720 -id:A058720 - cp's OEIS frontend

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

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

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

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

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

A058721 Number of simple matroids on n labeled points.

Original entry on oeis.org

1, 2, 7, 49, 733, 29760, 9000402

Offset: 2

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 A058720.

A058722 Number of simple matroids of rank 4 on n labeled points.

Original entry on oeis.org

1, 16, 337, 18700, 7642631

Offset: 4

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

Column k=4 of A058720.

A100728 Number of rank-(n-2) simple matroids on S_n.

Original entry on oeis.org

1, 31, 337, 2570, 16865, 104858, 650761, 4145956, 27483392, 190522216, 1382087111, 10478149999, 82860356456, 682066659044, 5832719543338, 51724107920729, 474869705028520, 4506715494154371, 44152005320340946

Offset: 4

Ralf Stephan, Nov 29 2004

nonn

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. [See Lemma 2.2(iii).]
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g. [See Lemma 2.2(iii).]

Cf. A000110 (Bell numbers). Diagonal of A058720.

a(n) = Bell(n+1) - (n^2+n+4)*2^(n-2) + n*(n+1)*(3*n^2-n+10)/24.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A058721 Number of simple matroids on n labeled points.

Views

Author

Keywords

Links

Crossrefs

A058722 Number of simple matroids of rank 4 on n labeled points.

Views

Author

Keywords

Links

Crossrefs

A100728 Number of rank-(n-2) simple matroids on S_n.

Views

Author

Keywords

Links

Crossrefs

Formula