A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870

Offset

1,3

Comments

A multiset is normal if it spans an initial interval of positive integers.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Example

The a(3) = 8 multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,2}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@Table[Select[mps[m], Intersection@@#=={}&], {m, allnorm[n]}]], {n, 6}]

Prog

(PARI)
P(n, k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n, k)={my(p=P(n, k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p, i, y) + polcoef(q, i, y)*sum(j=1, n\i, (-1)^j*binomial(k, j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021

Crossrefs

Cf. A007716, A255903, A255906, A317073, A281116, A317077, A317532, A317533.

Cf. A317748, A317751, A317755, A317757, A317776.

Sequence in context: A037539 A037483 A188708 * A273497 A024106 A176626

Adjacent sequences: A317749 A317750 A317751 * A317753 A317754 A317755

Keyword

nonn

Author

Gus Wiseman, Aug 06 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021

Status

approved