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A309876
Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes with at least one (possibly empty) hyperedge; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
3
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 33, 33, 5, 1, 1, 6, 155, 2135, 155, 6, 1, 1, 7, 1043, 7013319, 7013319, 1043, 7, 1, 1, 8, 12345, 1788782616655, 29281354514767167, 1788782616655, 12345, 8, 1
OFFSET
0,5
COMMENTS
A hypergraph is called k-uniform if all hyperedges have the same cardinality k.
T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
LINKS
Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.
Wikipedia, Hypergraph
FORMULA
T(n,k) = A309865(n,k) - 1 = A309858(n,k) - 1.
T(n,k) = T(n,n-k) for 0 <= k <= n.
EXAMPLE
T(3,0) = 1: {{}}.
T(3,1) = 3: {1}, {1,2}, {1,2,3}.
T(3,2) = 3: {12}, {12,13}, {12,13,23}.
T(3,3) = 1: {123}.
(Non-isomorphic representatives of the hypergraphs are given.)
Triangle T(n,k) begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 10, 4, 1;
1, 5, 33, 33, 5, 1;
1, 6, 155, 2135, 155, 6, 1;
1, 7, 1043, 7013319, 7013319, 1043, 7, 1;
...
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)-1)
end:
seq(seq(T(n, k), k=0..n), n=0..9);
CROSSREFS
Columns k=0-1 give: A000012, A001477.
Row sums give A309868.
T(2n,n) gives A328157.
Sequence in context: A283113 A123610 A209631 * A059922 A229556 A159623
KEYWORD
nonn,tabl
AUTHOR
Peter Dolland and Alois P. Heinz, Aug 21 2019
STATUS
approved