A309858
Number A(n,k) of k-uniform hypergraphs on n unlabeled nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
2, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 1, 1, 4, 5, 2, 1, 1, 1, 2, 11, 6, 2, 1, 1, 1, 1, 5, 34, 7, 2, 1, 1, 1, 1, 2, 34, 156, 8, 2, 1, 1, 1, 1, 1, 6, 2136, 1044, 9, 2, 1, 1, 1, 1, 1, 2, 156, 7013320, 12346, 10, 2, 1, 1, 1, 1, 1, 1, 7, 7013320, 1788782616656, 274668, 11, 2
Offset: 0
Square array A(n,k) begins:
2, 1, 1, 1, 1, 1, 1, 1, ...
2, 2, 1, 1, 1, 1, 1, 1, ...
2, 3, 2, 1, 1, 1, 1, 1, ...
2, 4, 4, 2, 1, 1, 1, 1, ...
2, 5, 11, 5, 2, 1, 1, 1, ...
2, 6, 34, 34, 6, 2, 1, 1, ...
2, 7, 156, 2136, 156, 7, 2, 1, ...
2, 8, 1044, 7013320, 7013320, 1044, 8, 2, ...
Columns k=0..10 give:
A007395,
A000027,
A000088,
A000665,
A051240,
A051249,
A309860,
A309861,
A309862,
A309863,
A309864.
-
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)):
A:= proc(n, k) option remember; `if`(k>n, 1,
`if`(k>n-k, A(n, n-k), b(n$2, [], k)))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
permcount(v)={my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
T(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!} \\ Andrew Howroyd, Aug 22 2019
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.
Original entry on oeis.org
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
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;
...
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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);
A309868
Number of uniform hypergraphs on n unlabeled nodes with at least one (possibly empty) hyperedge.
Original entry on oeis.org
1, 2, 4, 8, 20, 78, 2459, 14028740, 29284932080025185, 468863491068204454517854447175206, 1994324729204021501147398087008429477142243091610827370319897909501551
Offset: 0
Non-isomorphic representatives of the a(3) = 8 uniform hypergraphs on 3 unlabeled nodes with at least one hyperedge: {{}}, {1}, {1,2}, {1,2,3}, {12}, {12,13}, {12,13,23}, {123}.
-
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) T(n, k):=`if`(k>n-k, T(n, n-k), b(n$2, [], k)) end:
a:= n-> add(T(n, k)-1, k=0..n):
seq(a(n), n=0..10);
A309895
Number of floor(n/2)-uniform hypergraphs on n unlabeled nodes.
Original entry on oeis.org
2, 2, 3, 4, 11, 34, 2136, 7013320, 29281354514767168, 234431745534048922731115555415680, 1994324729203114587259985605157804740271034553359179870979936357974016
Offset: 0
Showing 1-4 of 4 results.
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