%I #5 Aug 13 2019 13:20:00

%S 1,1,1,2,1,2,1,3,2,2,1,2,1,2,2,5,1,2,1,2,2,2,1,3,2,2,3,2,1,5,1,7,2,2,

%T 2,9,1,2,2,3,1,5,1,2,2,2,1,5,2,2,2,2,1,3,2,3,2,2,1,4,1,2,2,11,2,5,1,2,

%U 2,5,1,12,1,2,2,2,2,5,1,5,5,2,1,4,2,2

%N Number of factorizations of n into factors > 1 whose dual is a weak antichain.

%C 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.

%C A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.

%e The a(36) = 9 factorizations:

%e (36)

%e (4*9)

%e (6*6)

%e (2*18)

%e (3*12)

%e (2*2*9)

%e (2*3*6)

%e (3*3*4)

%e (2*2*3*3)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[facs[n],stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

%Y The T_0 case (where the dual is strict) is A316978.

%Y Set-systems whose dual is a weak antichain are A326968.

%Y Partitions whose dual is a weak antichain are A326978.

%Y The T_1 case (where the dual is a strict antichain) is A327012.

%Y Cf. A001055, A059523, A316978, A319639, A326965, A326966, A326969, A326970, A326971, A326976, A326977.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 13 2019