cp's OEIS Frontend

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

%I A327012 #4 Aug 15 2019 07:30:03
%S A327012 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,5,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,7,1,1,
%T A327012 1,7,1,1,1,3,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,11,1,1,1,2,
%U A327012 1,1,1,12,1,1,2,2,1,1,1,5,5,1,1,2,1,1
%N A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.
%C A327012 Differs from A322453 at 36, 72, 100, ...
%C A327012 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 A327012 An antichain is a set of multisets, none of which is a submultiset of any other.
%e A327012 The a(72) = 12 factorizations:
%e A327012   (8*9)
%e A327012   (3*24)
%e A327012   (4*18)
%e A327012   (2*4*9)
%e A327012   (3*3*8)
%e A327012   (3*4*6)
%e A327012   (2*2*18)
%e A327012   (2*3*12)
%e A327012   (2*2*2*9)
%e A327012   (2*2*3*6)
%e A327012   (2*3*3*4)
%e A327012   (2*2*2*3*3)
%t A327012 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A327012 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A327012 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A327012 submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
%t A327012 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A327012 Table[Length[Select[facs[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]
%Y A327012 Set-systems whose dual is a (strict) antichain are A326965.
%Y A327012 The version where the dual is a weak antichain is A326975.
%Y A327012 Partitions whose dual is a (strict) antichain are A326977.
%Y A327012 Cf. A001055, A316978, A326961, A326974, A326976, A326979.
%K A327012 nonn
%O A327012 1,4
%A A327012 _Gus Wiseman_, Aug 13 2019