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A059087 Triangle T(n,m) of number of labeled n-node T_0-hypergraphs with m distinct hyperedges (empty hyperedge excluded), m=0,1,...,2^n-1. 7
1, 1, 1, 0, 2, 3, 1, 0, 0, 12, 32, 35, 21, 7, 1, 0, 0, 12, 256, 1155, 2877, 4963, 6429, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 0, 0, 0, 1120, 19040, 140616, 686476, 2565260, 7824375, 20110025, 44322135, 84658665, 141115975, 206252025, 265182375 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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} s(n, i)*binomial(2^i-1, m), where s(n, i) are Stirling numbers of the first kind.
Also T(n, m) = (1/m!)*Sum_{i=0..m+1} s(m+1, i)*fallfac(2^(i-1), n). E.g.f: Sum((1+x)^(2^n-1)*log(1+y)^n/n!, n=0..infinity). - Vladeta Jovovic, May 19 2004
EXAMPLE
Triangle starts:
[1],
[1,1],
[0,2,3,1],
[0,0,12,32,35,21,7,1],
...;
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 - 1, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n - 1}] // Flatten (* Jean-François Alcover, Sep 02 2016 *)
CROSSREFS
Sequence in context: A010341 A072772 A124314 * A353493 A321932 A321933
KEYWORD
easy,nonn,tabf
AUTHOR
Goran Kilibarda, Vladeta Jovovic, Dec 27 2000
STATUS
approved

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Last modified March 1 19:16 EST 2024. Contains 370443 sequences. (Running on oeis4.)