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

0,9

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)

Links

Table of n, a(n) for n=0..44.

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.

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)

Example

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;
  ...

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).

Sequence in context: A348600 A200545 A294522 * A281891 A124539 A351703

Adjacent sequences: A058707 A058708 A058709 * A058711 A058712 A058713

Keyword

nonn,tabl,nice

Author

N. J. A. Sloane, Dec 31 2000

Extensions

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

Status

approved