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A059084 Triangle T(n,m) of number of labeled n-node T_0-hypergraphs with m distinct hyperedges (empty hyperedge included), m=0,1,...,2^n. 14
1, 1, 1, 2, 1, 0, 2, 5, 4, 1, 0, 0, 12, 44, 67, 56, 28, 8, 1, 0, 0, 12, 268, 1411, 4032, 7840, 11392, 12864, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 1120, 20160, 159656, 827092, 3251736, 10389635, 27934400, 64432160, 128980800, 225774640 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A hypergraph is a T_0 hypergraph if for every two distinct nodes there exists a hyperedge containing one but not the other node.
LINKS
FORMULA
T(n,m) = Sum_{i=0..n} Stirling1(n,i) * binomial(2^i,m).
T(n,m) = A181230(n,m) / m!.
From Vladeta Jovovic, May 19 2004: (Start)
T(n, m) = (1/m!)*Sum_{i=0..m} s(m, i)*fallfac(2^i, n).
E.g.f.: Sum_{n>=0} (1+x)^(2^n)*log(1+y)^n/n!. (End)
EXAMPLE
Triangle begins:
m 0 1 2 3 4 5 6 7 8 sums A059085(n)
n
0 1 1 2
1 1 2 1 4
1 0 2 5 4 1 12
2 0 0 12 44 67 56 28 8 1 216
There are 12 labeled 3-node T_0-hypergraphs with 2 distinct hyperedges: {{3},{2}}, {{3},{2,3}}, {{2},{2,3}}, {{3},{1}}, {{3},{1,3}}, {{2},{1}}, {{2,3},{1,3}}, {{2},{1,2}}, {{2,3},{1,2}}, {{1},{1,3}}, {{1},{1,2}}, {{1,3},{1,2}}.
MATHEMATICA
T[n_, m_] := Sum[StirlingS1[n, i] Binomial[2^i, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n}] // Flatten (* Jean-François Alcover, Sep 02 2016 *)
CROSSREFS
Cf. A088309.
Sequence in context: A355650 A292892 A074142 * A246117 A295688 A355610
KEYWORD
easy,nonn,tabf
AUTHOR
Goran Kilibarda, Vladeta Jovovic, Dec 27 2000
STATUS
approved

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Last modified February 24 11:15 EST 2024. Contains 370303 sequences. (Running on oeis4.)