OFFSET
1,6
COMMENTS
From Mats Granvik, May 25 2017: (Start)
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
(2*6) (3*8) (5*6) (4*8) (4*9)
(3*4) (4*6) (6*5) (8*4) (6*6)
(4*3) (6*4) (10*3) (16*2) (9*4)
(6*2) (8*3) (15*2) (2*16) (12*3)
(12*2) (2*15) (2*2*2*4) (18*2)
(2*12) (3*10) (2*2*4*2) (2*18)
(2*2*2*3) (2*4*2*2) (3*12)
(2*2*3*2) (4*2*2*2) (2*2*3*3)
(2*3*2*2) (2*3*2*3)
(3*2*2*2) (2*3*3*2)
(3*2*2*3)
(3*2*3*2)
(3*3*2*2)
(End)
LINKS
Mats Granvik, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A002033.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019
MATHEMATICA
(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n], EvenQ@*Length]], {n, 100}] (* Gus Wiseman, Jan 04 2021 *)
CROSSREFS
The odd version is A174726.
The unordered version is A339846.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
KEYWORD
nonn
AUTHOR
Mats Granvik, Mar 28 2010
STATUS
approved