A348613 Number of non-alternating ordered factorizations of n.

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

Offset: 1

Author

Gus Wiseman, Nov 03 2021

Keywords

• nonn

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.

Examples

			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
		

Links

• Wikipedia, Alternating permutation

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.

Programs

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}]