OFFSET
1,6
COMMENTS
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
EXAMPLE
The a(n) factorizations for n = 2, 6, 16, 12, 30, 24, 36, 48, 60:
2 6 16 12 30 24 36 48 60
2*3 2*8 2*6 5*6 3*8 4*9 6*8 2*30
2*2*4 3*4 2*15 4*6 2*18 2*24 3*20
2*2*3 3*10 2*12 3*12 3*16 4*15
2*3*5 2*2*6 2*2*9 4*12 5*12
2*3*4 2*3*6 2*3*8 6*10
3*3*4 2*4*6 2*5*6
2*2*3*3 3*4*4 3*4*5
2*2*12 2*2*15
2*2*3*4 2*3*10
2*2*3*5
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]!={}&]], {n, 100}]
CROSSREFS
The version for partitions is A325534.
The inseparable version is A333487.
The version for multisets with prescribed multiplicities is A335127.
Factorizations are A001055.
Anti-run compositions are A003242.
Inseparable partitions are A325535.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 03 2020
STATUS
approved