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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). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

LINKS

Table of n, a(n) for n=2..29.

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

FORMULA

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)

EXAMPLE

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;

  ...

CROSSREFS

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).

Sequence in context: A143213 A172377 A156587 * A015116 A322220 A174790

Adjacent sequences:  A058717 A058718 A058719 * A058721 A058722 A058723

KEYWORD

nonn,tabl,nice,more

AUTHOR

N. J. A. Sloane, Dec 31 2000

STATUS

approved

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Last modified January 21 11:11 EST 2020. Contains 331105 sequences. (Running on oeis4.)