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A349059 Number of weakly alternating ordered factorizations of n. 17
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, 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, 28, 3, 11, 1, 8, 3, 11, 1, 56, 1, 3, 8, 8, 3, 11, 1, 38, 8, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
LINKS
FORMULA
a(2^n) = A349052(n).
EXAMPLE
The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
(2) (4) (6) (8) (12) (24) (30)
(2*2) (2*3) (2*4) (2*6) (3*8) (5*6)
(3*2) (4*2) (3*4) (4*6) (6*5)
(2*2*2) (4*3) (6*4) (10*3)
(6*2) (8*3) (15*2)
(2*2*3) (12*2) (2*15)
(2*3*2) (2*12) (3*10)
(3*2*2) (2*2*6) (2*5*3)
(2*4*3) (3*2*5)
(2*6*2) (3*5*2)
(3*2*4) (5*2*3)
(3*4*2)
(4*2*3)
(6*2*2)
(2*2*2*3)
(2*2*3*2)
(2*3*2*2)
(3*2*2*2)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n], whkQ[#]||whkQ[-#]&]], {n, 100}]
CROSSREFS
The strong version for compositions is A025047, also A025048, A025049.
The strong case is A348610, complement A348613.
The version for compositions is A349052, complement A349053.
As compositions these are ranked by the complement of A349057.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating permutations of prime factors, w/ twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 = factorizations w/ alternating permutation, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
A349800 = weakly but not strongly alternating compositions, ranked A349799.
Sequence in context: A296119 A300836 A118314 * A002033 A074206 A173801
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2021
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)