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

Offset

0,4

Comments

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

Equals row sums of triangle A180338. - Gary W. Adamson, Aug 28 2010

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Crossrefs

Column k = 2 of A058669.

The triangle A340264 without the main diagonal provides a refinement of this sequence.

Cf. A180338, A005465.

Sequence in context: A038748 A099455 A102053 * A246417 A110310 A054493

Adjacent sequences: A058678 A058679 A058680 * A058682 A058683 A058684

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Dec 30 2000

Extensions

More terms from James A. Sellers, Jan 03 2001

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

Status

approved