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A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 5
2, 2, 2, 2, 3, 2, 2, 4, 4, 2, 2, 5, 11, 5, 2, 2, 6, 34, 34, 6, 2, 2, 7, 156, 2136, 156, 7, 2, 2, 8, 1044, 7013320, 7013320, 1044, 8, 2, 2, 9, 12346, 1788782616656, 29281354514767168, 1788782616656, 12346, 9, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n.  T(n,k) = 1 for k>n.

See A000088 and A000665 for more references.

LINKS

Alois P. Heinz, Rows n = 0..14, flattened

Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.

Wikipedia, Hypergraph

FORMULA

T(n,k) = T(n,n-k) for 0 <= k <= n.

EXAMPLE

Triangle T(n,k) begins:

  2;

  2, 2;

  2, 3,    2;

  2, 4,    4,       2;

  2, 5,   11,       5,       2;

  2, 6,   34,      34,       6,    2;

  2, 7,  156,    2136,     156,    7, 2;

  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)):

T:= proc(n, k) option remember; `if`(k>n-k,

      T(n, n-k), b(n$2, [], k))

    end:

seq(seq(T(n, k), k=0..n), n=0..9);

CROSSREFS

Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864.

Cf. A309858 (the same as square array).

Sequence in context: A308622 A198897 A201375 * A128764 A324818 A233417

Adjacent sequences:  A309862 A309863 A309864 * A309866 A309867 A309868

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 20 2019

STATUS

approved

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Last modified January 24 04:35 EST 2020. Contains 331183 sequences. (Running on oeis4.)