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A330235
Number of fully chiral factorizations of n.
10
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, 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, 4, 0, 0, 1, 16, 1, 0, 4, 4, 0, 0, 1, 12, 5, 0, 1, 4, 0, 0
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.
Sequence in context: A272894 A268387 A136566 * A048983 A301505 A301504
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2019
STATUS
approved