A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset

1,32

Comments

First differs from A348381 at a(216) = 4, A348381(216) = 3.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

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 sets.

Links

Table of n, a(n) for n=1..87.

Wikipedia, Alternating permutation

Formula

a(2^n) = A344654(n).

Example

The a(n) factorizations for n = 96, 192, 2160, 576:
  2*2*2*12      3*4*4*4         3*3*3*80       4*4*4*9
  2*2*2*2*6     2*2*2*24        6*6*6*10       2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*270      2*2*2*2*36
                2*2*2*2*2*6     2*3*3*3*40     2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*135    2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*3*45   2*2*2*2*2*18
                                2*2*2*2*5*27   2*2*2*2*3*12
                                2*2*2*2*9*15   2*2*2*2*2*2*9
                                               2*2*2*2*2*3*6
                                               2*2*2*2*2*2*3*3

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Function[f, Select[Permutations[f], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]=={}]]], {n, 100}]

Crossrefs

Positions of nonzero terms are A046099.

Partitions of this type are counted by A344654, ranked by A344653.

Partitions not of this type are counted by A344740, ranked by A344742.

The complement is counted by A347050, without twins A348379.

The version for compositions is A348377.

The version allowing twins is A348380.

The inseparable case is A348381.

A001055 counts factorizations, strict A045778, ordered A074206.

A001250 counts alternating permutations of sets.

A025047 counts alternating or wiggly compositions, ranked by A345167.

A339846 counts even-length factorizations.

A339890 counts odd-length factorizations.

A347438 counts factorizations with alternating product 1, additive A119620.

A348610 counts alternating ordered factorizations.

Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.

Sequence in context: A056626 A366123 A290081 * A348381 A010103 A086078

Adjacent sequences: A347703 A347704 A347705 * A347707 A347708 A347709

Keyword

nonn

Author

Gus Wiseman, Oct 28 2021

Status

approved