A330235 Number of fully chiral factorizations of n.

1, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 4, 1, 0, 0, 5, 1, 4, 1, 4, 0, 0, 1, 7, 2, 0, 3, 4, 1, 0, 1, 7, 0, 0, 0, 4, 1, 0, 0, 7, 1, 0, 1, 4, 4, 0, 1, 12, 2, 4, 0, 4, 1, 7, 0, 7, 0, 0, 1, 4, 1, 0, 4, 11, 0, 0, 1, 4, 0, 0, 1, 16, 1, 0, 4, 4, 0, 0, 1, 12, 5, 0, 1, 4, 0, 0

Offset

1,4

Comments

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative. A factorization is fully chiral if taking the multiset of prime indices of each factor gives a fully chiral multiset of multisets.

Links

Table of n, a(n) for n=1..86.

Example

The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
  ()  (4)    (8)      (12)     (16)       (24)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (3*16)
                               (2*2*2*2)  (2*2*6)    (4*12)
                                          (2*3*4)    (2*3*8)
                                          (2*2*2*3)  (2*4*6)
                                                     (3*4*4)
                                                     (2*2*12)
                                                     (2*2*2*6)
                                                     (2*2*3*4)
                                                     (2*2*2*2*3)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[facs[n], Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]], {n, 100}]

Crossrefs

The costrict (or T_0) version is A316978.

The achiral version is A330234.

Planted achiral trees are A003238.

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.

MM-numbers of fully chiral multisets of multisets are A330236.

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

Sequence in context: A272894 A268387 A136566 * A048983 A301505 A301504

Adjacent sequences: A330232 A330233 A330234 * A330236 A330237 A330238

Keyword

nonn

Author

Gus Wiseman, Dec 08 2019

Status

approved