

A330236


MMnumbers of fully chiral multisets of multisets.


14



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, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83
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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 MMnumber 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 MMnumber 78 is {{},{1},{1,2}}.


LINKS



FORMULA



EXAMPLE

The sequence of all fully chiral multisets of multisets together with their MMnumbers 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

Costrict (or T_0) factorizations are A316978.
BIInumbers of fully chiral setsystems are A330226.
Nonisomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering setsystems are A330229.
Fully chiral factorizations are A330235.


KEYWORD

nonn


AUTHOR



STATUS

approved



