cp's OEIS Frontend

A330235 Number of fully chiral factorizations of n.

%I A330235 #5 Dec 09 2019 07:28:30
%S A330235 1,1,1,2,1,0,1,3,2,0,1,4,1,0,0,5,1,4,1,4,0,0,1,7,2,0,3,4,1,0,1,7,0,0,
%T A330235 0,4,1,0,0,7,1,0,1,4,4,0,1,12,2,4,0,4,1,7,0,7,0,0,1,4,1,0,4,11,0,0,1,
%U A330235 4,0,0,1,16,1,0,4,4,0,0,1,12,5,0,1,4,0,0
%N A330235 Number of fully chiral factorizations of n.
%C A330235 A multiset of multisets is fully chiral every permutation of the vertices gives a different representative. A factorization is fully chiral if taking the multiset of prime indices of each factor gives a fully chiral multiset of multisets.
%e A330235 The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
%e A330235   ()  (4)    (8)      (12)     (16)       (24)       (48)
%e A330235       (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (6*8)
%e A330235              (2*2*2)  (3*4)    (4*4)      (4*6)      (2*24)
%e A330235                       (2*2*3)  (2*2*4)    (2*12)     (3*16)
%e A330235                                (2*2*2*2)  (2*2*6)    (4*12)
%e A330235                                           (2*3*4)    (2*3*8)
%e A330235                                           (2*2*2*3)  (2*4*6)
%e A330235                                                      (3*4*4)
%e A330235                                                      (2*2*12)
%e A330235                                                      (2*2*2*6)
%e A330235                                                      (2*2*3*4)
%e A330235                                                      (2*2*2*2*3)
%t A330235 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330235 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330235 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330235 Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]],{n,100}]
%Y A330235 The costrict (or T_0) version is A316978.
%Y A330235 The achiral version is A330234.
%Y A330235 Planted achiral trees are A003238.
%Y A330235 BII-numbers of fully chiral set-systems are A330226.
%Y A330235 Non-isomorphic fully chiral multiset partitions are A330227.
%Y A330235 Full chiral partitions are A330228.
%Y A330235 Fully chiral covering set-systems are A330229.
%Y A330235 MM-numbers of fully chiral multisets of multisets are A330236.
%Y A330235 Cf. A001055, A007716, A083323, A112798, A317533, A330098, A330223, A330224, A330232.
%K A330235 nonn
%O A330235 1,4
%A A330235 _Gus Wiseman_, Dec 08 2019