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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).

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

2,5

LINKS

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, 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.]

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, 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

Row sums give A058721.

Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).

KEYWORD

nonn,tabl,nice,more

AUTHOR

STATUS

approved

URL: https://oeis.org/A058720