A321188 Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n).

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 11, 0, 0, 0, 4, 0, 0, 0, 1

Offset: 1

Author

Gus Wiseman, Oct 29 2018

Keywords

• nonn
• more

Comments

A set system is a finite set of finite nonempty sets.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(36) = 4 set systems with no singletons whose multiset union is {1,1,2,2,3,4}:
  {{1,2},{1,2,3,4}}
  {{1,2,3},{1,2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}
		

Crossrefs

Cf. A000070, A000296, A000569, A050326, A056239, A112798, A283877, A292444, A305936, A306005, A318285, A318361, A320922, A320923, A320924.

Programs

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]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[hyp[nrmptn[n]]],{n,30}]