OFFSET
1,32
COMMENTS
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.
FORMULA
EXAMPLE
The a(n) factorizations for n = 96, 192, 384, 576:
2*2*2*12 3*4*4*4 4*4*4*6 4*4*4*9
2*2*2*2*6 2*2*2*24 2*2*2*48 2*2*2*72
2*2*2*2*2*3 2*2*2*2*12 2*2*2*2*24 2*2*2*2*36
2*2*2*2*2*6 2*2*2*2*3*8 2*2*2*2*4*9
2*2*2*2*3*4 2*2*2*2*4*6 2*2*2*2*6*6
2*2*2*2*2*2*3 2*2*2*2*2*12 2*2*2*2*2*18
2*2*2*2*2*2*6 2*2*2*2*3*12
2*2*2*2*2*3*4 2*2*2*2*2*2*9
2*2*2*2*2*2*2*3 2*2*2*2*2*3*6
2*2*2*2*2*2*3*3
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], !MatchQ[#, {x_, x_}]&&Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]=={}&]], {n, 100}]
CROSSREFS
Positions of nonzero terms are A046099.
Allowing twins gives A333487.
A001250 counts alternating permutations of sets.
A025047 counts alternating or wiggly compositions.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 30 2021
STATUS
approved