A174726 a(n) = (A002033(n-1) - A008683(n))/2.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1

Offset: 1

Author

Mats Granvik, Mar 28 2010

Keywords

• nonn

Comments

a(n) is the number of permutation matrices with a negative contribution to the determinant that is the Möbius function. See A174725 for how the determinant is defined. - Mats Granvik, May 26 2017

From Gus Wiseman, Jan 04 2021: (Start)

Also the number of ordered factorizations of n into an odd number of factors > 1. The unordered case is A339890. For example, the a(n) factorizations for n = 8, 12, 24, 30, 32, 36 are:

(8) (12) (24) (30) (32) (36)

(2*2*2) (2*2*3) (2*2*6) (2*3*5) (2*2*8) (2*2*9)

(2*3*2) (2*3*4) (2*5*3) (2*4*4) (2*3*6)

(3*2*2) (2*4*3) (3*2*5) (2*8*2) (2*6*3)

(2*6*2) (3*5*2) (4*2*4) (2*9*2)

(3*2*4) (5*2*3) (4*4*2) (3*2*6)

(3*4*2) (5*3*2) (8*2*2) (3*3*4)

(4*2*3) (2*2*2*2*2) (3*4*3)

(4*3*2) (3*6*2)

(6*2*2) (4*3*3)

(6*2*3)

(6*3*2)

(9*2*2)

(End)

Links

• Mats Granvik, Table of n, a(n) for n = 1..10000

Crossrefs

The even version is A174725.

The unordered case is A339890, with even version A339846.

A001055 counts factorizations, with strict case A045778.

A074206 counts ordered factorizations, with strict case A254578.

A251683 counts ordered factorizations by product and length.

A340102 counts odd-length factorizations into odd factors.

Other cases of odd length:

- A024429 counts set partitions of odd length.

- A027193 counts partitions of odd length.

- A067659 counts strict partitions of odd length.

- A089677 counts ordered set partitions of odd length.

- A166444 counts compositions of odd length.

- A332304 counts strict compositions of odd length.

Cf. A002033, A024430, A027187, A050320, A052841, A058695, A160786, A316439.

Programs

Mathematica

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],OddQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

Formula

a(n) = (A002033(n-1) - A008683(n))/2. - Mats Granvik, May 26 2017

For n > 0, a(n) + A174725(n) = A074206(n). - Gus Wiseman, Jan 04 2021