A330236 MM-numbers of fully chiral multisets of multisets.

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

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}}.

Links

Table of n, a(n) for n=1..67.

Formula

Numbers n such that A330098(n) = A303975(n)!.

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

Costrict (or T_0) factorizations are A316978.

BII-numbers of fully chiral set-systems are A330226.

Non-isomorphic fully chiral multiset partitions are A330227.

Full chiral partitions are A330228.

Fully chiral covering set-systems are A330229.

Fully chiral factorizations are A330235.

Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.

Sequence in context: A058226 A172974 A066255 * A302593 A129304 A337149

Adjacent sequences: A330233 A330234 A330235 * A330237 A330238 A330239

Keyword

nonn

Author

Gus Wiseman, Dec 10 2019

Status

approved