Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #4 Dec 09 2019 07:27:57
%S 1,1,2,3,5,6,9,12,18,25,33,45,61,80,106,140,176,232,293,381,476,615,
%T 764,975,1191,1511,1849,2322,2812,3517,4231,5240,6297,7736,9260,11315,
%U 13468,16378,19485,23531,27851,33525,39585,47389,55844,66517,78169,92810
%N Number of fully chiral integer partitions of n.
%C 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.
%e The a(1) = 1 through a(7) = 12 partitions:
%e (1) (2) (3) (4) (5) (33) (7)
%e (11) (21) (22) (41) (42) (43)
%e (111) (31) (221) (51) (322)
%e (211) (311) (222) (331)
%e (1111) (2111) (411) (421)
%e (11111) (2211) (511)
%e (3111) (2221)
%e (21111) (4111)
%e (111111) (22111)
%e (31111)
%e (211111)
%e (1111111)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t Table[Length[Select[IntegerPartitions[n],Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]],{n,0,15}]
%Y The Heinz numbers of these partitions are given by A330236.
%Y Costrict (or T_0) partitions are A319564.
%Y Achiral partitions are A330224.
%Y BII-numbers of fully chiral set-systems are A330226.
%Y Non-isomorphic, fully chiral multiset partitions are A330227.
%Y Fully chiral covering set-systems are A330229.
%Y Fully chiral factorizations are A330235.
%Y Cf. A001055, A007716, A322847, A330098, A330223.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 08 2019