A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

2, 2, 2, 2, 3, 2, 2, 4, 4, 2, 2, 5, 11, 5, 2, 2, 6, 34, 34, 6, 2, 2, 7, 156, 2136, 156, 7, 2, 2, 8, 1044, 7013320, 7013320, 1044, 8, 2, 2, 9, 12346, 1788782616656, 29281354514767168, 1788782616656, 12346, 9, 2

Offset: 0

Author

Alois P. Heinz, Aug 20 2019

Keywords

• nonn
• tabl

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 1 for k>n.

See A000088 and A000665 for more references.

Examples

			Triangle T(n,k) begins:
  2;
  2, 2;
  2, 3,    2;
  2, 4,    4,       2;
  2, 5,   11,       5,       2;
  2, 6,   34,      34,       6,    2;
  2, 7,  156,    2136,     156,    7, 2;
  2, 8, 1044, 7013320, 7013320, 1044, 8, 2;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..14, flattened
• Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.
• Wikipedia, Hypergraph

Crossrefs

Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864.

Cf. A309858 (the same as square array).

Programs

Maple

g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
     [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
     /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
     /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
    `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
     /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
T:= proc(n, k) option remember; `if`(k>n-k,
      T(n, n-k), b(n$2, [], k))
    end:
seq(seq(T(n, k), k=0..n), n=0..9);

Formula

T(n,k) = T(n,n-k) for 0 <= k <= n.