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A058669 Triangle T(n,k) read by rows, giving number of matroids of rank k on n labeled points (n >= 0, 0 <= k <= n). 5
1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 36, 15, 1, 1, 31, 171, 171, 31, 1, 1, 63, 813, 2053, 813, 63, 1, 1, 127, 4012, 33442, 33442, 4012, 127, 1, 1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1, 1, 511 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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.
FORMULA
From Petros Hadjicostas, Oct 10 2019: (Start)
T(n,0) = 1 for n >= 0.
T(n,1) = 2^n - 1 for n >= 1. [Dukes (2004), Theorem 2.1 (ii).
T(n,2) = Bell(n+1) - 2^n = A000110(n+1) - A000079(n) for n >= 2. [Dukes (2004), Theorem 2.1 (ii).]
T(n,k) = Sum_{m = k..n} binomial(n,m) * A058711(m,k) for n >= k. [Dukes (2004), see the equations before Theorem 2.1.]
(End)
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 36, 15, 1;
1, 31, 171, 171, 31, 1;
1, 63, 813, 2053, 813, 63, 1;
1, 127, 4012, 33442, 33442, 4012, 127, 1;
1, 255, 20891, 1022217, 8520812, 1022217, 20891, 255, 1;
...
CROSSREFS
Row sums give A058673.
Columns include (truncated versions of) A000012 (k=0), A000225 (k=1), A058681 (k=2), A058687 (k=3).
Sequence in context: A359985 A022166 A141689 * A057004 A059328 A174387
KEYWORD
nonn,nice,tabl,more
AUTHOR
N. J. A. Sloane, Dec 30 2000
STATUS
approved

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Last modified April 23 20:33 EDT 2024. Contains 371916 sequences. (Running on oeis4.)