A350139 - OEIS

%I #10 Jan 07 2022 15:54:49

%S 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 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 0,0,2,0,20,0,0,0,0,0,2,0,10,0,0,0,12,0

%N Number of non-weakly alternating ordered factorizations of n.

%C The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.

%C An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.

%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) = A349053(n).

%e The a(n) ordered factorizations for n = 24, 36, 48, 60:

%e   (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)

%e   (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)

%e            (2*3*3*2)  (6*4*2)    (5*4*3)

%e            (3*2*2*3)  (8*3*2)    (6*5*2)

%e                       (2*2*3*4)  (10*3*2)

%e                       (2*3*4*2)  (2*3*10)

%e                       (2*4*3*2)  (2*2*3*5)

%e                       (3*2*2*4)  (2*3*5*2)

%e                       (4*2*2*3)  (2*5*3*2)

%e                       (4*3*2*2)  (3*2*2*5)

%e                                  (5*2*2*3)

%e                                  (5*3*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 Positions of nonzero terms are A122181.

%Y The strong version for compositions is A345192, ranked by A345168.

%Y The strong case is A348613, complement A348610.

%Y The version for compositions is A349053, complement A349052.

%Y As compositions with ones allowed these are ranked by A349057.

%Y The complement is counted by A349059.

%Y A001055 counts factorizations, strict A045778, ordered A074206.

%Y A001250 counts alternating permutations, complement A348615.

%Y A025047 counts weakly alternating compositions, ranked by A345167.

%Y A335434 counts separable factorizations, complement A333487.

%Y A345164 counts alternating perms of prime factors, with twins A344606.

%Y A345170 counts partitions with an alternating permutation.

%Y A348379 counts factorizations w/ alternating perm, complement A348380.

%Y A348611 counts anti-run ordered factorizations, complement A348616.

%Y A349060 counts weakly alternating partitions, complement A349061.

%Y Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

%K nonn

%O 1,24

%A _Gus Wiseman_, Dec 24 2021