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 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. 15
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A000088 and A000665 for more references. LINKS Alois P. Heinz, Antidiagonals n = 0..20, flattened Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989. Wikipedia, Hypergraph FORMULA A(n,k) = A(n,n-k) for 0 <= k <= n. A(n,k) - A(n-1,k) = A301922(n,k) for n,k >= 1. EXAMPLE 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, ... 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)): 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); PROG (PARI) 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 CROSSREFS Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864. Cf. A301922, A309865 (the same as triangle). Sequence in context: A023518 A326194 A331251 * A022921 A080763 A245920 Adjacent sequences: A309855 A309856 A309857 * A309859 A309860 A309861 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 20 2019 STATUS approved

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Last modified September 11 06:30 EDT 2024. Contains 375814 sequences. (Running on oeis4.)