OFFSET
0,3
COMMENTS
A multiset partition is fully chiral if every permutation of the vertices gives a different representative. An integer partition is fully chiral if taking the multiset of prime indices of each part gives a fully chiral multiset of multisets.
EXAMPLE
The a(1) = 1 through a(7) = 12 partitions:
(1) (2) (3) (4) (5) (33) (7)
(11) (21) (22) (41) (42) (43)
(111) (31) (221) (51) (322)
(211) (311) (222) (331)
(1111) (2111) (411) (421)
(11111) (2211) (511)
(3111) (2221)
(21111) (4111)
(111111) (22111)
(31111)
(211111)
(1111111)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[IntegerPartitions[n], Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]], {n, 0, 15}]
CROSSREFS
The Heinz numbers of these partitions are given by A330236.
Costrict (or T_0) partitions are A319564.
Achiral partitions are A330224.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2019
STATUS
approved