OFFSET
1,6
COMMENTS
First differs from A335434 at a(216) = 27, A335434(216) = 28. Also differs from A335434 at a(270) = 19, A335434(270) = 20.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All of the counted factorizations are separable (A335434).
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 {1..n}.
LINKS
FORMULA
a(2^n) = A345170(n).
EXAMPLE
The a(270) = 19 factorizations:
(2*3*3*15) (2*3*45) (2*135) (270)
(2*3*5*9) (2*5*27) (3*90)
(3*3*5*6) (2*9*15) (5*54)
(3*3*30) (6*45)
(3*5*18) (9*30)
(3*6*15) (10*27)
(3*9*10) (15*18)
(5*6*9)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[facs[n], Select[Permutations[#], wigQ]!={}&]], {n, 100}]
CROSSREFS
The case with twins is A347050.
The ordered version is A348610.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2021
STATUS
approved