OFFSET
1,4
COMMENTS
The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.
A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.
EXAMPLE
The a(36) = 9 factorizations:
(36)
(4*9)
(6*6)
(2*18)
(3*12)
(2*2*9)
(2*3*6)
(3*3*4)
(2*2*3*3)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[facs[n], stableQ[dual[primeMS/@#], submultQ]&]], {n, 100}]
CROSSREFS
The T_0 case (where the dual is strict) is A316978.
Set-systems whose dual is a weak antichain are A326968.
Partitions whose dual is a weak antichain are A326978.
The T_1 case (where the dual is a strict antichain) is A327012.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 13 2019
STATUS
approved