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A350139 Number of non-weakly alternating ordered factorizations of n.

%I A350139 #10 Jan 07 2022 15:54:49
%S A350139 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,
%T A350139 0,4,0,0,0,2,0,2,0,0,0,0,0,10,0,0,0,0,0,2,0,2,0,0,0,12,0,0,0,4,0,2,0,
%U A350139 0,0,2,0,20,0,0,0,0,0,2,0,10,0,0,0,12,0
%N A350139 Number of non-weakly alternating ordered factorizations of n.
%C A350139 The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.
%C A350139 An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.
%C A350139 We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%F A350139 a(2^n) = A349053(n).
%e A350139 The a(n) ordered factorizations for n = 24, 36, 48, 60:
%e A350139   (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
%e A350139   (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
%e A350139            (2*3*3*2)  (6*4*2)    (5*4*3)
%e A350139            (3*2*2*3)  (8*3*2)    (6*5*2)
%e A350139                       (2*2*3*4)  (10*3*2)
%e A350139                       (2*3*4*2)  (2*3*10)
%e A350139                       (2*4*3*2)  (2*2*3*5)
%e A350139                       (3*2*2*4)  (2*3*5*2)
%e A350139                       (4*2*2*3)  (2*5*3*2)
%e A350139                       (4*3*2*2)  (3*2*2*5)
%e A350139                                  (5*2*2*3)
%e A350139                                  (5*3*2*2)
%t A350139 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A350139 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A350139 Table[Length[Select[Join@@Permutations/@facs[n],!whkQ[#]&&!whkQ[-#]&]],{n,100}]
%Y A350139 Positions of nonzero terms are A122181.
%Y A350139 The strong version for compositions is A345192, ranked by A345168.
%Y A350139 The strong case is A348613, complement A348610.
%Y A350139 The version for compositions is A349053, complement A349052.
%Y A350139 As compositions with ones allowed these are ranked by A349057.
%Y A350139 The complement is counted by A349059.
%Y A350139 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A350139 A001250 counts alternating permutations, complement A348615.
%Y A350139 A025047 counts weakly alternating compositions, ranked by A345167.
%Y A350139 A335434 counts separable factorizations, complement A333487.
%Y A350139 A345164 counts alternating perms of prime factors, with twins A344606.
%Y A350139 A345170 counts partitions with an alternating permutation.
%Y A350139 A348379 counts factorizations w/ alternating perm, complement A348380.
%Y A350139 A348611 counts anti-run ordered factorizations, complement A348616.
%Y A350139 A349060 counts weakly alternating partitions, complement A349061.
%Y A350139 Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.
%K A350139 nonn
%O A350139 1,24
%A A350139 _Gus Wiseman_, Dec 24 2021