This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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;
...
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);
Comments