|
| |
|
|
A348379
|
|
Number of factorizations of n with an alternating permutation.
|
|
27
|
|
|
|
1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All of the counted factorizations are separable (A335434).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
The a(270) = 19 factorizations:
(2*3*3*15) (2*3*45) (2*135) (270)
(2*3*5*9) (2*5*27) (3*90)
(3*3*5*6) (2*9*15) (5*54)
(3*3*30) (6*45)
(3*5*18) (9*30)
(3*6*15) (10*27)
(3*9*10) (15*18)
(5*6*9)
|
|
|
MATHEMATICA
|
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=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[facs[n], Select[Permutations[#], wigQ]!={}&]], {n, 100}]
|
|
|
CROSSREFS
|
Partitions not of this type are counted by A345165, ranked by A345171.
Twins and partitions of this type are counted by A344740, ranked by A344742.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A056986, A325534, A335452, A347437, A347438, A347439, A347442, A347456, A347463, A348381, A348383, A348609.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
