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

Author

Gus Wiseman, Aug 25 2018

Keywords

• nonn

Comments

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

Examples

			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}}
		

Links

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

Crossrefs

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

Programs

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

Extensions

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