A330459 Number of set partitions of set-systems with total sum n.

1, 1, 1, 4, 6, 11, 26, 42, 78, 148, 280, 481, 867, 1569, 2742, 4933, 8493, 14857, 25925, 44877, 77022, 132511, 226449, 385396, 657314, 1111115, 1875708, 3157379, 5309439, 8885889, 14861478, 24760339, 41162971, 68328959, 113099231, 186926116, 308230044

Offset: 0

Author

Gus Wiseman, Dec 17 2019

Keywords

• nonn

Comments

Number of sets of disjoint nonempty sets of nonempty sets of positive integers with total sum n.

Examples

			The a(6) = 26 partitions:
  ((6))  ((15))      ((123))          ((1)(2)(12))
         ((24))      ((1)(14))        ((1))((2)(12))
         ((1)(5))    ((1)(23))        ((12))((1)(2))
         ((2)(4))    ((2)(13))        ((2))((1)(12))
         ((1))((5))  ((3)(12))        ((1))((2))((12))
         ((2))((4))  ((1))((14))
                     ((1))((23))
                     ((1)(2)(3))
                     ((2))((13))
                     ((3))((12))
                     ((1))((2)(3))
                     ((2))((1)(3))
                     ((3))((1)(2))
                     ((1))((2))((3))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A007713, A050342, A050343, A279375, A279785, A283877, A294617, A330460, A330462, A323787-A323795, A330452-A330459.

Programs

Mathematica

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[UnsameQ@@Join@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

PARI

\\ here L is A000009 and BellP is A000110 as series.
L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(c=L(n), b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c, k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ Andrew Howroyd, Dec 29 2019

Formula

a(n) = Sum_k A330462(n,k) * A000110(k).

Extensions

Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019