login
A330228
Number of fully chiral integer partitions of n.
12
1, 1, 2, 3, 5, 6, 9, 12, 18, 25, 33, 45, 61, 80, 106, 140, 176, 232, 293, 381, 476, 615, 764, 975, 1191, 1511, 1849, 2322, 2812, 3517, 4231, 5240, 6297, 7736, 9260, 11315, 13468, 16378, 19485, 23531, 27851, 33525, 39585, 47389, 55844, 66517, 78169, 92810
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.
Sequence in context: A244747 A241742 A212584 * A166048 A240306 A094873
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 08 2019
STATUS
approved