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

0,1

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.

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

Formula

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

Example

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;
  ...

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);

Crossrefs

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

Cf. A309858 (the same as square array).

Sequence in context: A308622 A198897 A201375 * A128764 A324818 A233417

Adjacent sequences: A309862 A309863 A309864 * A309866 A309867 A309868

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 20 2019

Status

approved