A330236 - OEIS

%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