cp's OEIS Frontend

A330224 Number of achiral integer partitions of n.

%I A330224 #11 Jun 26 2020 11:53:32
%S A330224 1,1,2,3,5,7,11,13,18,21,30,32,43,46,57,64,79,83,103,107,130,141,162,
%T A330224 171,205,214,245,258,297,307,357,373,423,441,493,513,591,607,674,702,
%U A330224 790,817,917,938,1040,1078,1186,1216,1362,1395,1534,1580,1738,1779,1956
%N A330224 Number of achiral integer partitions of n.
%C A330224 A multiset of multisets is achiral if it is not changed by any permutation of the vertices. An integer partition is achiral if taking the multiset of prime indices of each part gives an achiral multiset of multisets.
%e A330224 The a(1) = 1 through a(7) = 13 partitions:
%e A330224   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A330224        (11)  (21)   (22)    (32)     (33)      (52)
%e A330224              (111)  (31)    (41)     (42)      (61)
%e A330224                     (211)   (221)    (51)      (331)
%e A330224                     (1111)  (311)    (222)     (421)
%e A330224                             (2111)   (321)     (511)
%e A330224                             (11111)  (411)     (2221)
%e A330224                                      (2211)    (3211)
%e A330224                                      (3111)    (4111)
%e A330224                                      (21111)   (22111)
%e A330224                                      (111111)  (31111)
%e A330224                                                (211111)
%e A330224                                                (1111111)
%t A330224 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330224 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330224 Table[Length[Select[IntegerPartitions[n],Length[graprms[primeMS/@#]]==1&]],{n,0,30}]
%Y A330224 The fully-chiral version is A330228.
%Y A330224 The Heinz numbers of these partitions are given by A330232.
%Y A330224 Achiral set-systems are counted by A083323.
%Y A330224 BII-numbers of achiral set-systems are A330217.
%Y A330224 Non-isomorphic achiral multiset partitions are A330223.
%Y A330224 Achiral factorizations are A330234.
%Y A330224 Cf. A001055, A056239, A112798, A319564, A330098, A330230.
%K A330224 nonn
%O A330224 0,3
%A A330224 _Gus Wiseman_, Dec 08 2019
%E A330224 More terms from _Jinyuan Wang_, Jun 26 2020