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%I #7 Dec 10 2019 12:12:15
%S 1,2,3,4,5,6,7,8,9,10,11,12,14,16,17,18,19,20,21,22,23,24,25,27,28,31,
%T 32,34,35,36,37,38,39,40,41,42,44,45,46,48,49,50,53,54,56,57,59,61,62,
%U 63,64,65,67,68,69,70,71,72,74,75,76,77,78,80,81,82,83
%N MM-numbers of fully chiral multisets of multisets.
%C A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.
%C 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}}.
%F Numbers n such that A330098(n) = A303975(n)!.
%e The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
%e 1: 18: {}{1}{1} 37: {112} 57: {1}{111}
%e 2: {} 19: {111} 38: {}{111} 59: {7}
%e 3: {1} 20: {}{}{2} 39: {1}{12} 61: {122}
%e 4: {}{} 21: {1}{11} 40: {}{}{}{2} 62: {}{5}
%e 5: {2} 22: {}{3} 41: {6} 63: {1}{1}{11}
%e 6: {}{1} 23: {22} 42: {}{1}{11} 64: {}{}{}{}{}{}
%e 7: {11} 24: {}{}{}{1} 44: {}{}{3} 65: {2}{12}
%e 8: {}{}{} 25: {2}{2} 45: {1}{1}{2} 67: {8}
%e 9: {1}{1} 27: {1}{1}{1} 46: {}{22} 68: {}{}{4}
%e 10: {}{2} 28: {}{}{11} 48: {}{}{}{}{1} 69: {1}{22}
%e 11: {3} 31: {5} 49: {11}{11} 70: {}{2}{11}
%e 12: {}{}{1} 32: {}{}{}{}{} 50: {}{2}{2} 71: {113}
%e 14: {}{11} 34: {}{4} 53: {1111} 72: {}{}{}{1}{1}
%e 16: {}{}{}{} 35: {2}{11} 54: {}{1}{1}{1} 74: {}{112}
%e 17: {4} 36: {}{}{1}{1} 56: {}{}{}{11} 75: {1}{2}{2}
%e The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.
%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 Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]
%Y Costrict (or T_0) factorizations are A316978.
%Y BII-numbers of fully chiral set-systems are A330226.
%Y Non-isomorphic fully chiral multiset partitions are A330227.
%Y Full chiral partitions are A330228.
%Y Fully chiral covering set-systems are A330229.
%Y Fully chiral factorizations are A330235.
%Y Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 10 2019