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A309858
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Number A(n,k) of k-uniform hypergraphs on n unlabeled nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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15
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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
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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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.
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EXAMPLE
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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, ...
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MAPLE
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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)):
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);
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PROG
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(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(#L<k, listput(L, #L))); Vec(L)}
can(v, f)={my(d=1, u=v); while(d>0, 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
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CROSSREFS
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Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864.
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KEYWORD
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AUTHOR
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STATUS
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approved
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