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

1,16

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.

Links

Table of n, a(n) for n=1..40.

Example

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}}

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}]

Crossrefs

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

Sequence in context: A222399 A222519 A128131 * A322076 A115633 A115713

Adjacent sequences: A321185 A321186 A321187 * A321189 A321190 A321191

Keyword

nonn,more

Author

Gus Wiseman, Oct 29 2018

Status

approved