A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

1, 1, 2, 7, 13, 34, 81, 175, 403, 890, 1977, 4262, 9356, 19963, 42573, 90865, 191206, 401803, 837898, 1744231, 3607504, 7436628, 15254309, 31185686, 63552725, 128963236, 260933000, 526140540, 1057927323, 2120500885, 4239012067, 8449746787, 16799938614

Offset

0,3

Comments

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

Links

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

Formula

a(n) = Sum_{0 <= k <= n} A330463(n,k) * A000110(k).

Example

The a(4) = 13 partitions:
  ((4))  ((22))  ((31))      ((211))      ((1111))
                 ((1)(3))    ((1)(21))    ((1)(111))
                 ((1))((3))  ((2)(11))    ((1))((111))
                             ((1))((21))
                             ((2))((11))

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], UnsameQ@@Join@@#&]], {n, 0, 10}]

Prog

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

Crossrefs

Cf. A001970, A007713, A050343, A063834, A089259, A261049, A271619, A279375, A294617, A318565, A323787-A323795, A330452-A330459, A330460.

Sequence in context: A079119 A309368 A051748 * A086904 A358906 A360869

Adjacent sequences: A330449 A330450 A330451 * A330453 A330454 A330455

Keyword

nonn

Author

Gus Wiseman, Dec 16 2019

Extensions

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

Status

approved