|
| |
|
|
A348613
|
|
Number of non-alternating ordered factorizations of n.
|
|
24
|
|
|
|
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 8, 1, 0, 1, 2, 0, 2, 0, 9, 0, 0, 0, 11, 0, 0, 0, 8, 0, 2, 0, 2, 2, 0, 0, 25, 1, 2, 0, 2, 0, 8, 0, 8, 0, 0, 0, 16, 0, 0, 2, 20, 0, 2, 0, 2, 0, 2, 0, 43, 0, 0, 2, 2, 0, 2, 0, 25, 4, 0, 0, 16, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,12
|
|
|
COMMENTS
|
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
|
|
|
LINKS
|
Table of n, a(n) for n=1..85.
Wikipedia, Alternating permutation
|
|
|
EXAMPLE
|
The a(n) ordered factorizations for n = 4, 12, 16, 24, 32, 36:
2*2 2*2*3 4*4 2*2*6 2*2*8 6*6
3*2*2 2*2*4 2*3*4 2*4*4 2*2*9
4*2*2 4*3*2 4*4*2 2*3*6
2*2*2*2 6*2*2 8*2*2 3*3*4
2*2*2*3 2*2*2*4 4*3*3
2*2*3*2 2*2*4*2 6*3*2
2*3*2*2 2*4*2*2 9*2*2
3*2*2*2 4*2*2*2 2*2*3*3
2*2*2*2*2 2*3*3*2
3*2*2*3
3*3*2*2
|
|
|
MATHEMATICA
|
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[ordfacs[n], !wigQ[#]&]], {n, 100}]
|
|
|
CROSSREFS
|
The complementary additive version is A025047, ranked by A345167.
The additive version is A345192, ranked by A345168, without twins A348377.
The complement is counted by A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions without an alternating permutation, ranked by A345171.
A345170 counts partitions with an alternating permutation, ranked by A345172.
A348379 counts factorizations w/ an alternating permutation, with twins A347050.
A348380 counts factorizations w/o an alternating permutation, w/o twins A347706.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A056986, A344614, A344653, A344654, A344740, A347437, A347438, A347463, A348381, A348609.
Sequence in context: A221381 A100951 A348616 * A285182 A190608 A011991
Adjacent sequences: A348610 A348611 A348612 * A348614 A348615 A348616
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Gus Wiseman, Nov 03 2021
|
|
|
STATUS
|
approved
|
| |
|
|
