OFFSET
1,32
COMMENTS
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.
LINKS
FORMULA
a(2^n) = A344654(n).
EXAMPLE
The a(n) factorizations for n = 96, 192, 2160, 576:
2*2*2*12 3*4*4*4 3*3*3*80 4*4*4*9
2*2*2*2*6 2*2*2*24 6*6*6*10 2*2*2*72
2*2*2*2*2*3 2*2*2*2*12 2*2*2*270 2*2*2*2*36
2*2*2*2*2*6 2*3*3*3*40 2*2*2*2*4*9
2*2*2*2*3*4 2*2*2*2*135 2*2*2*2*6*6
2*2*2*2*2*2*3 2*2*2*2*3*45 2*2*2*2*2*18
2*2*2*2*5*27 2*2*2*2*3*12
2*2*2*2*9*15 2*2*2*2*2*2*9
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], Function[f, Select[Permutations[f], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]=={}]]], {n, 100}]
CROSSREFS
Positions of nonzero terms are A046099.
The version for compositions is A348377.
The version allowing twins is A348380.
The inseparable case is A348381.
A001250 counts alternating permutations of sets.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348610 counts alternating ordered factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2021
STATUS
approved