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.

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

Offset: 0

Author

Goran Kilibarda, Vladeta Jovovic, Dec 27 2000

Keywords

• easy
• nonn
• tabf

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.

Examples

			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}}.
		

Links

• V. Jovovic, Illustration of initial terms of A059087, A059088

Crossrefs

Cf. A059084, A059085, A059086, A059088, A059089.

Programs

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 *)

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