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

Author

N. J. A. Sloane, Dec 31 2000

Keywords

• nonn
• tabl
• nice

Comments

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)

Examples

			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;
  ...
		

Links

• 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.

Crossrefs

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).

Formula

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)

Extensions

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