cp's OEIS Frontend

A349059 Number of weakly alternating ordered factorizations of n.

%I A349059 #6 Dec 10 2021 11:13:40
%S A349059 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 A349059 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 A349059 28,3,11,1,8,3,11,1,56,1,3,8,8,3,11,1,38,8,3
%N A349059 Number of weakly alternating ordered factorizations of n.
%C A349059 An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%C A349059 We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%F A349059 a(2^n) = A349052(n).
%e A349059 The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
%e A349059   (2)  (4)    (6)    (8)      (12)     (24)       (30)
%e A349059        (2*2)  (2*3)  (2*4)    (2*6)    (3*8)      (5*6)
%e A349059               (3*2)  (4*2)    (3*4)    (4*6)      (6*5)
%e A349059                      (2*2*2)  (4*3)    (6*4)      (10*3)
%e A349059                               (6*2)    (8*3)      (15*2)
%e A349059                               (2*2*3)  (12*2)     (2*15)
%e A349059                               (2*3*2)  (2*12)     (3*10)
%e A349059                               (3*2*2)  (2*2*6)    (2*5*3)
%e A349059                                        (2*4*3)    (3*2*5)
%e A349059                                        (2*6*2)    (3*5*2)
%e A349059                                        (3*2*4)    (5*2*3)
%e A349059                                        (3*4*2)
%e A349059                                        (4*2*3)
%e A349059                                        (6*2*2)
%e A349059                                        (2*2*2*3)
%e A349059                                        (2*2*3*2)
%e A349059                                        (2*3*2*2)
%e A349059                                        (3*2*2*2)
%t A349059 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A349059 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]], {m,1,Length[y]-1}];
%t A349059 Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]],{n,100}]
%Y A349059 The strong version for compositions is A025047, also A025048, A025049.
%Y A349059 The strong case is A348610, complement A348613.
%Y A349059 The version for compositions is A349052, complement A349053.
%Y A349059 As compositions these are ranked by the complement of A349057.
%Y A349059 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A349059 A001250 counts alternating permutations, complement A348615.
%Y A349059 A335434 counts separable factorizations, complement A333487.
%Y A349059 A345164 counts alternating permutations of prime factors, w/ twins A344606.
%Y A349059 A345170 counts partitions with an alternating permutation.
%Y A349059 A348379 = factorizations w/ alternating permutation, complement A348380.
%Y A349059 A348611 counts anti-run ordered factorizations, complement A348616.
%Y A349059 A349060 counts weakly alternating partitions, complement A349061.
%Y A349059 A349800 = weakly but not strongly alternating compositions, ranked A349799.
%Y A349059 Cf. A003242, A122181, A138364, A339846, A339890, A345165, A345167, A345194, A347050, A347438, A347463, A347706.
%K A349059 nonn
%O A349059 1,4
%A A349059 _Gus Wiseman_, Dec 04 2021