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A330232 MM-numbers of achiral multisets of multisets.

%I A330232 #8 Mar 26 2020 20:41:50
%S A330232 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A330232 27,28,29,30,31,32,33,34,36,38,40,41,42,43,44,46,47,48,49,50,51,52,53,
%U A330232 54,55,56,57,58,59,60,62,63,64,66,67,68,72,73,76,79,80
%N A330232 MM-numbers of achiral multisets of multisets.
%C A330232 First differs from A322554 in lacking 141.
%C A330232 A multiset of multisets is achiral if it is not changed by any permutation of the vertices.
%C A330232 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%e A330232 The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
%e A330232   35: {{2},{1,1}}
%e A330232   37: {{1,1,2}}
%e A330232   39: {{1},{1,2}}
%e A330232   45: {{1},{1},{2}}
%e A330232   61: {{1,2,2}}
%e A330232   65: {{2},{1,2}}
%e A330232   69: {{1},{2,2}}
%e A330232   70: {{},{2},{1,1}}
%e A330232   71: {{1,1,3}}
%e A330232   74: {{},{1,1,2}}
%e A330232   75: {{1},{2},{2}}
%e A330232   77: {{1,1},{3}}
%e A330232   78: {{},{1},{1,2}}
%e A330232   87: {{1},{1,3}}
%e A330232   89: {{1,1,1,2}}
%e A330232   90: {{},{1},{1},{2}}
%t A330232 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330232 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]]
%t A330232 Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==1&]
%Y A330232 The fully-chiral version is A330236.
%Y A330232 Achiral set-systems are counted by A083323.
%Y A330232 MG-numbers of planted achiral trees are A214577.
%Y A330232 MM-weight is A302242.
%Y A330232 MM-numbers of costrict (or T_0) multisets of multisets are A322847.
%Y A330232 BII-numbers of achiral set-systems are A330217.
%Y A330232 Non-isomorphic achiral multiset partitions are A330223.
%Y A330232 Achiral integer partitions are counted by A330224.
%Y A330232 Achiral factorizations are A330234.
%Y A330232 Cf. A001055, A003238, A007716, A056239, A112798, A303975, A330098, A330230, A330233.
%K A330232 nonn
%O A330232 1,2
%A A330232 _Gus Wiseman_, Dec 08 2019