A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

1, 2, 6, 20, 74, 311, 1401, 6913, 36376, 205421, 1228288, 7786802, 51937607, 364250763, 2673314121, 20504809133, 163844631872, 1361874185139, 11748149246269, 105029750531640, 971403871953460, 9282643841237360, 91519776792040324, 929892817423282068, 9725646244888190337

Offset

1,2

Comments

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Example

The a(4) = 20 sets of sets:
  {{1,2,3,4}}
  {{1},{1,2,3}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{3,4}}
  {{1},{3},{1,2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Prog

(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p, n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

Cf. A007716, A049311, A050326, A116540, A283877, A292432, A292444, A318361, A318369, A318370.

Sequence in context: A150158 A034010 A135588 * A293492 A150159 A150160

Adjacent sequences: A318399 A318400 A318401 * A318403 A318404 A318405

Keyword

nonn

Author

Gus Wiseman, Aug 25 2018

Extensions

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

Status

approved