%I #6 Dec 10 2021 11:13:40
%S 1,1,1,2,1,3,1,4,2,3,1,8,1,3,3,8,1,8,1,8,3,3,1,18,2,3,4,8,1,11,1,16,3,
%T 3,3,22,1,3,3,18,1,11,1,8,8,3,1,38,2,8,3,8,1,18,3,18,3,3,1,32,1,3,8,
%U 28,3,11,1,8,3,11,1,56,1,3,8,8,3,11,1,38,8,3
%N Number of weakly alternating ordered factorizations of n.
%C An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%C We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%F a(2^n) = A349052(n).
%e The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
%e (2) (4) (6) (8) (12) (24) (30)
%e (2*2) (2*3) (2*4) (2*6) (3*8) (5*6)
%e (3*2) (4*2) (3*4) (4*6) (6*5)
%e (2*2*2) (4*3) (6*4) (10*3)
%e (6*2) (8*3) (15*2)
%e (2*2*3) (12*2) (2*15)
%e (2*3*2) (2*12) (3*10)
%e (3*2*2) (2*2*6) (2*5*3)
%e (2*4*3) (3*2*5)
%e (2*6*2) (3*5*2)
%e (3*2*4) (5*2*3)
%e (3*4*2)
%e (4*2*3)
%e (6*2*2)
%e (2*2*2*3)
%e (2*2*3*2)
%e (2*3*2*2)
%e (3*2*2*2)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]], {m,1,Length[y]-1}];
%t Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]],{n,100}]
%Y The strong version for compositions is A025047, also A025048, A025049.
%Y The strong case is A348610, complement A348613.
%Y The version for compositions is A349052, complement A349053.
%Y As compositions these are ranked by the complement of A349057.
%Y A001055 counts factorizations, strict A045778, ordered A074206.
%Y A001250 counts alternating permutations, complement A348615.
%Y A335434 counts separable factorizations, complement A333487.
%Y A345164 counts alternating permutations of prime factors, w/ twins A344606.
%Y A345170 counts partitions with an alternating permutation.
%Y A348379 = factorizations w/ alternating permutation, complement A348380.
%Y A348611 counts anti-run ordered factorizations, complement A348616.
%Y A349060 counts weakly alternating partitions, complement A349061.
%Y A349800 = weakly but not strongly alternating compositions, ranked A349799.
%Y Cf. A003242, A122181, A138364, A339846, A339890, A345165, A345167, A345194, A347050, A347438, A347463, A347706.
%K nonn
%O 1,4
%A _Gus Wiseman_, Dec 04 2021