OFFSET
1,4
COMMENTS
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.
EXAMPLE
The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
() (4) (8) (12) (16) (24) (48)
(2*2) (2*4) (2*6) (2*8) (3*8) (6*8)
(2*2*2) (3*4) (4*4) (4*6) (2*24)
(2*2*3) (2*2*4) (2*12) (3*16)
(2*2*2*2) (2*2*6) (4*12)
(2*3*4) (2*3*8)
(2*2*2*3) (2*4*6)
(3*4*4)
(2*2*12)
(2*2*2*6)
(2*2*3*4)
(2*2*2*2*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]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[facs[n], Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]], {n, 100}]
CROSSREFS
The costrict (or T_0) version is A316978.
The achiral version is A330234.
Planted achiral trees are A003238.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
MM-numbers of fully chiral multisets of multisets are A330236.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2019
STATUS
approved