OFFSET
1,2
COMMENTS
A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.
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}}.
EXAMPLE
The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
1: 18: {}{1}{1} 37: {112} 57: {1}{111}
2: {} 19: {111} 38: {}{111} 59: {7}
3: {1} 20: {}{}{2} 39: {1}{12} 61: {122}
4: {}{} 21: {1}{11} 40: {}{}{}{2} 62: {}{5}
5: {2} 22: {}{3} 41: {6} 63: {1}{1}{11}
6: {}{1} 23: {22} 42: {}{1}{11} 64: {}{}{}{}{}{}
7: {11} 24: {}{}{}{1} 44: {}{}{3} 65: {2}{12}
8: {}{}{} 25: {2}{2} 45: {1}{1}{2} 67: {8}
9: {1}{1} 27: {1}{1}{1} 46: {}{22} 68: {}{}{4}
10: {}{2} 28: {}{}{11} 48: {}{}{}{}{1} 69: {1}{22}
11: {3} 31: {5} 49: {11}{11} 70: {}{2}{11}
12: {}{}{1} 32: {}{}{}{}{} 50: {}{2}{2} 71: {113}
14: {}{11} 34: {}{4} 53: {1111} 72: {}{}{}{1}{1}
16: {}{}{}{} 35: {2}{11} 54: {}{1}{1}{1} 74: {}{112}
17: {4} 36: {}{}{1}{1} 56: {}{}{}{11} 75: {1}{2}{2}
The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.
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]}]];
Select[Range[100], Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 10 2019
STATUS
approved