A053534 Triangle T(n,k) giving number of pairwise non-isomorphic (i.e., unlabeled) matroids of rank k on n points (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 23, 38, 23, 6, 1, 1, 7, 37, 108, 108, 37, 7, 1, 1, 8, 58, 325, 940, 325, 58, 8, 1, 1, 9, 87, 1275, 190214, 190214, 1275, 87, 9, 1

Offset: 0

Author

N. J. A. Sloane, Dec 30 2000

Keywords

• nonn
• tabl
• nice

Examples

			The triangle, transposed, begins:
k...n=0...n=1...n=2...n=3...n=4...n=5...n=6...n=7...n=8...n=9...
0.|.1.....1.....1.....1.....1.....1.....1.....1.....1.......1.....
1.|.......1.....2.....3.....4.....5.....6.....7.....8.......9.....
2.|.............1.....3.....7....13....23....37....58......87.....
3.|...................1.....4....13....38...108...325....1275.....
4.|.........................1.....5....23...108...940..190214.....
5.|...............................1.....6....37...325..190214.....
6.|.....................................1.....7....58....1275.....
7.|...........................................1.....8......87.....
8.|.................................................1.......9.....
9.|.........................................................1.....
Sum.1.....2.....4.....8....17....38....98...306..1724..383172
		

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.
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
• Index entries for sequences related to matroids

Crossrefs

Row sums give A055545.

Columns include (truncated versions of) A000012 (k=0), A000027 (k=1), A058682 (k=2), A058693 (k=3).

Cf. A000041, A058669.

Formula

From Petros Hadjicostas, Oct 10 2019: (Start)

T(n,0) = 1 for n >= 0.

T(n,1) = n for n >= 1.

T(n,2) = -n + Sum_{k = 1..n} p(k) for n >= 2, where p(k) = A000041(k). [Dukes (2004), Theorem 2.1.] (End)

Extensions

More terms from Jonathan Vos Post, Feb 14 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar and Max Alekseyev