A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.

0, 0, 1, 3, 7, 13, 23, 37, 58, 87, 128, 183, 259, 359, 493, 668, 898, 1194, 1578, 2067, 2693, 3484, 4485, 5739, 7313, 9270, 11705, 14714, 18431, 22995, 28598, 35439, 43787, 53929, 66238, 81120, 99096, 120732, 146746, 177930, 215267, 259849, 313022, 376282

Offset

0,4

Comments

Number of non-isomorphic rank-2 matroids over S_n.

Starting (1, 3, 7, 13, ...) = row sums of triangle A171239. - Gary W. Adamson, Dec 05 2009

References

Acketa, Dragan M. "On the enumeration of matroids of rank-2." Zbornik radova Prirodnomatematickog fakulteta-Univerzitet u Novom Sadu 8 (1978): 83-90. - N. J. A. Sloane, Dec 04 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Muniru A Asiru)

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.

Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.

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

Formula

G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Aug 10 2018

Maple

a:= proc(n) option remember; `if`(n<2, 0,
      combinat[numbpart](n)+a(n-1)-1)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 10 2019

Mathematica

With[{s = PartitionsP /@ Range[0, 40]}, MapIndexed[Total@ Take[s, First@ #2] - First@ #2 &, s]] (* Michael De Vlieger, Sep 04 2017 *)
With[{nn=50}, #[[2]]-#[[1]]&/@Thread[{Range[0, nn], Accumulate[PartitionsP[Range[0, nn]]]}]]-1 (* Harvey P. Dale, Sep 05 2023 *)

Prog

(GAP) List([1..41], n->Sum([1..n-1], k->NrPartitions(k)-1)); # Muniru A Asiru, Aug 10 2018

Crossrefs

Column k=2 of A053534.

Cf. A000041, A000065 (first differences), A000070.

Cf. A171239. - Gary W. Adamson, Dec 05 2009

Sequence in context: A164787 A131205 A256309 * A081995 A291141 A053599

Adjacent sequences: A058679 A058680 A058681 * A058683 A058684 A058685

Keyword

nonn

Author

N. J. A. Sloane, Dec 30 2000

Extensions

Name clarified by Ilya Gutkovskiy, Aug 10 2018

Status

approved