cp's OEIS Frontend

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.

%I A309876 #22 Oct 05 2019 21:06:43
%S A309876 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,
%T A309876 7,1043,7013319,7013319,1043,7,1,1,8,12345,1788782616655,
%U A309876 29281354514767167,1788782616655,12345,8,1
%N 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.
%C A309876 A hypergraph is called k-uniform if all hyperedges have the same cardinality k.
%C A309876 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.
%H A309876 Alois P. Heinz, <a href="/A309876/b309876.txt">Rows n = 0..14, flattened</a>
%H A309876 Jianguo Qian, <a href="https://doi.org/10.1016/j.disc.2014.03.005">Enumeration of unlabeled uniform hypergraphs</a>, Discrete Math. 326 (2014), 66--74. MR3188989.
%H A309876 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%F A309876 T(n,k) = A309865(n,k) - 1 = A309858(n,k) - 1.
%F A309876 T(n,k) = T(n,n-k) for 0 <= k <= n.
%e A309876 T(3,0) = 1: {{}}.
%e A309876 T(3,1) = 3: {1}, {1,2}, {1,2,3}.
%e A309876 T(3,2) = 3: {12}, {12,13}, {12,13,23}.
%e A309876 T(3,3) = 1: {123}.
%e A309876 (Non-isomorphic representatives of the hypergraphs are given.)
%e A309876 Triangle T(n,k) begins:
%e A309876   1;
%e A309876   1, 1;
%e A309876   1, 2,    1;
%e A309876   1, 3,    3,       1;
%e A309876   1, 4,   10,       4,       1;
%e A309876   1, 5,   33,      33,       5,    1;
%e A309876   1, 6,  155,    2135,     155,    6, 1;
%e A309876   1, 7, 1043, 7013319, 7013319, 1043, 7, 1;
%e A309876   ...
%p A309876 g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
%p A309876      [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
%p A309876 h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
%p A309876      /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
%p A309876      /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
%p A309876     `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
%p A309876 b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
%p A309876      /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
%p A309876 T:= proc(n, k) option remember; `if`(k>n-k,
%p A309876       T(n, n-k), b(n$2, [], k)-1)
%p A309876     end:
%p A309876 seq(seq(T(n, k), k=0..n), n=0..9);
%Y A309876 Columns k=0-1 give: A000012, A001477.
%Y A309876 Row sums give A309868.
%Y A309876 T(2n,n) gives A328157.
%Y A309876 Cf. A309858, A309865.
%K A309876 nonn,tabl
%O A309876 0,5
%A A309876 _Peter Dolland_ and _Alois P. Heinz_, Aug 21 2019