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A058681
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Number of matroids of rank 2 on n labeled points.
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5
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1, 7, 36, 171, 813, 4012, 20891, 115463, 677546, 4211549, 27640341, 190891130, 1382942161, 10480109379, 82864804268, 682076675087, 5832741942913, 51724157711084, 474869815108175, 4506715736350171, 44152005850890042
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| E.g.f.: exp(exp(z)-1)-1/2*exp(z)^2-1/2. - Thomas Wieder (wieder.thomas(AT)t-online.de), Nov 30 2004
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. - A. O. Munagi (amunagi(AT)yahoo.com), 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. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 20 2005
a(n) is the number of collections containing two or more nonempty subsets of {1,2,...n} that are pairwise disjoint. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Oct 10 2009]
Equals row sums of triangle A180338 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2010]
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REFERENCES
| A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2, (2005), 215-224.
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LINKS
| T. D. Noe, Table of n, a(n) for n=2..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
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
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FORMULA
| a(n) = B(n+1)-2^n, B = Bell numbers (A000110).
a(n)= Sum i=2...n,Binomial(n,i)*(B(i)-1), B=Bell numbers A000110 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Oct 10 2009]
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EXAMPLE
| a(3)=7 because the 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}. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Oct 10 2009]
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MAPLE
| series(exp(exp(z)-1)-1/2*exp(z)^2-1/2, z=0, 10);
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MATHEMATICA
| f[n_] := Sum[ StirlingS2[n, k+2], {k, 1, n}]; Table[ f[n], {n, 3, 23}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007
Table[BellB[n+1]-2^n, {n, 2, 30}] (* From Harvey P. Dale, Oct 13 2011 *)
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CROSSREFS
| A diagonal of A058669.
Cf. A180338 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 28 2010]
Sequence in context: A038748 A099455 A102053 * A110310 A054493 A037538
Adjacent sequences: A058678 A058679 A058680 * A058682 A058683 A058684
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 03 2001
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