A058669 -id:A058669 - cp's OEIS frontend

A058681 Number of matroids of rank 2 on n labeled points.

0, 0, 1, 7, 36, 171, 813, 4012, 20891, 115463, 677546, 4211549, 27640341, 190891130, 1382942161, 10480109379, 82864804268, 682076675087, 5832741942913, 51724157711084, 474869815108175, 4506715736350171, 44152005850890042, 445958869286416681, 4638590332213222137

Offset: 0

N. J. A. Sloane, Dec 30 2000

Number of partitions of {1, 2, ..., n+1} in which at least one block of each partition contains a pair of nonconsecutive integers. E.g., B(4)-2^3 = 7: there are 7 partitions of {1,2,3,4} in which some block contains a pair of nonconsecutive integers, namely 124/3, 134/2, 14/23, 13/24, 13/2/4, 14/2/3, 1/24/3. - Augustine O. Munagi, Mar 20 2005
Number of complementing systems of subsets of {0, 1, ..., p^(n+1) - 1} (p a prime) in which at least one member is not of the form {0, x, 2x, ..., (c-1)x} for positive integers x and c. E.g., B(4)-p^3 = 7: there are 7 complementing systems of subsets of {0, 1, ..., p^4-1} in which at least one member is not of the form {0, x, 2x, ..., (c-1)*x}. Number of complementing systems of subsets of {0, 1, ..., p^4 - 1} reduces to B(4) and number of ordered factorizations of p^4 is p^3. - Augustine O. Munagi, Mar 20 2005
a(n) is the number of collections containing two or more nonempty subsets of {1,2,...,n} that are pairwise disjoint. - Geoffrey Critzer, Oct 10 2009

			a(3) = 7 because there are 7 collections (having more than one element)of nonempty subsets of {1,2,3} that are pairwise disjoint: {1}{2}; {1}{3}; {1}{2,3}; {2}{3}; {2}{1,3}; {1,2}{3}; {1}{2}{3}. - _Geoffrey Critzer_, Oct 10 2009

T. D. Noe, Table of n, a(n) for n = 0..100
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, On the number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
I. J. Good, The number of hypotheses of independence for a random vector or for a multidimensional contingency table, and the Bell numbers, Iranian J. Science and Technology, 4, (1975), 77-83. [See Eq. (9), p. 80.]
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.
A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
Index entries for sequences related to matroids

Column k = 2 of A058669.
The triangle A340264 without the main diagonal provides a refinement of this sequence.
Cf. A005465.

egf := exp(x + exp(x) - 1) - exp(2*x); ser := series(egf, x, 24):
seq(simplify(n!*coeff(ser,x,n)), n=0..22); # Peter Luschny, Jan 08 2021

f[n_] := Sum[ StirlingS2[n + 1, k+2], {k, 1, n}]; Table[ f[n], {n, 0, 23}] (* Zerinvary Lajos, Mar 21 2007 *)
Table[BellB[n+1]-2^n,{n,0,30}] (* Harvey P. Dale, Oct 13 2011 *)

a(n) = sum(k=1, n, stirling(n+1, k+2, 2)); \\ Ruud H.G. van Tol, May 09 2024

my(x='x+O('x^33)); concat([0,0],Vec(serlaplace(exp(x + exp(x) - 1) - exp(2*x)))) \\ Joerg Arndt, May 10 2024

a(n) = B(n+1)-2^n, B = Bell numbers (A000110).
E.g.f.: d/dz (exp(exp(z)-1) - (1/2)*exp(2*z) - 1/2). - Thomas Wieder, Nov 30 2004
a(n) = Sum_{i=2..n} binomial(n,i)*(B(i)-1), B=Bell numbers A000110. - Geoffrey Critzer, Oct 10 2009
E.g.f.: exp(x + exp(x) - 1) - exp(2*x). - Peter Luschny, Jan 08 2021

More terms from James Sellers, Jan 03 2001

a(0) = a(1) = 0 prepended by Peter Luschny, Jan 08 2021

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).

Original entry on oeis.org

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

N. J. A. Sloane, Dec 30 2000

			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

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

Row sums give A055545.
Columns include (truncated versions of) A000012 (k=0), A000027 (k=1), A058682 (k=2), A058693 (k=3).
Cf. A000041, A058669.

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)

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

A058673 Number of matroids on n labeled points.

Original entry on oeis.org

1, 2, 5, 16, 68, 406, 3807, 75164, 10607540

Offset: 0

N. J. A. Sloane, Dec 30 2000

From Lorenzo Sauras Altuzarra, Aug 10 2023: (Start)
a(n) <= A014466(n).
a(n) <= A306020(n). (End)

			The 16 possible sets E such that ({1, 2, 3}, E) is a matroid:
  {{}}
  {{}, {1}}
  {{}, {2}}
  {{}, {3}}
  {{}, {1}, {2}}
  {{}, {1}, {3}}
  {{}, {2}, {3}}
  {{}, {1}, {2}, {3}}
  {{}, {1}, {2}, {1, 2}}
  {{}, {1}, {3}, {1, 3}}
  {{}, {2}, {3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

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.
S. C. Locke, Matroids
Index entries for sequences related to matroids

Row sums of A058669. Closely related to A114491.

Cf. A014466 (abstract simplicial complexes), A055545 (unlabeled matroids), A306020.

A256158 Triangular array of numbers of 2-polymatroids of rank k on n labeled points, for n>=0, 0<=k<=2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 3, 1, 1, 7, 29, 41, 29, 7, 1, 1, 15, 135, 477, 784, 477, 135, 15, 1, 1, 31, 642, 5957, 27375, 41695, 27375, 5957, 642, 31, 1, 1, 63, 3199, 87477, 1554077, 7109189, 21937982, 7109189, 1554077, 87477, 3199, 63, 1, 1, 127, 16879, 1604768, 189213842, 3559635761, 733133160992, 86322358307, 733133160992, 3559635761, 189213842, 1604768, 16879, 127, 1

Offset: 0

Max Alekseyev, Mar 16 2015

The rows are symmetric: a(n,k) = a(n,2n-k).

Starting with n=7, the rows are not unimodal.

			Triangle starts with:
n=0: 1
n=1: 1 1 1
n=2: 1 3 6 3 1
n=3: 1 7 29 41 29 7 1
n=4: 1 15 135 477 784 477 135 15 1
n=5: 1 31 642 5957 27375 41695 27375 5957 642 31 1
n=6: 1 63 3199 87477 1554077 7109189 21937982 7109189 1554077 87477 3199 63 1
n=7: 1 127 16879 1604768 189213842 3559635761 733133160992 86322358307 733133160992 3559635761 189213842 1604768 16879 127 1

Thomas J. Savitsky, Enumeration of 2-polymatroids on up to seven elements. SIAM J. Discrete Math., 28(4):1641-1650, 2014. arXiv:1401.8006

Cf. A256156, A256157, A256159, A058669

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A058681 Number of matroids of rank 2 on n labeled points.

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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).

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A058673 Number of matroids on n labeled points.

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A256158 Triangular array of numbers of 2-polymatroids of rank k on n labeled points, for n>=0, 0<=k<=2n.

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A058687 Number of matroids of rank 3 on n labeled points.

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