# [seqfan] Re: No. of sign-nonsingular matrices

Eric W. Weisstein eric at weisstein.com
Fri Mar 27 02:39:19 CET 2009

```On Thu, 26 Mar 2009, N. J. A. Sloane wrote:

> Seqfans,  I don't know if this is in the OEIS already.
> Could someone work out a few terms?
>
> I saw the definition in the paper "On sign-nonsingular matrices ...",
> by Brualdi and Shader, in the Victor Klee Festschrift volume (DIMACS/AMS Series,
> Vol 4)
>
> We are looking at n x n matrices A with entries {0,-1,+1}.
> We let |A| be the matrix obtained from A by replacing each
> entry by its absolute value.
>
> Then the question is, how many such matrices A have the property that
>
>         permanent( |A| ) =  | det A |  ?
>
> (Initially no group acts; later one could count them mod permutations
> of rows and columns, possibly allowing transposing.)

If I'm not mistaken, the first few terms are

2, 49, 8419

(might have the next one for you soon) which does not appear to be in
OEIS.

Cheers,
-Eric

E.g., here are the 2 for the 1x1 case:

{{{0}}, {{1}}}

and the 49 for the 2x2 case:

{{{-1, -1}, {0, -1}}, {{-1, -1}, {0,
0}}, {{-1, -1}, {1, -1}}, {{-1, -1}, {1, 0}}, {{-1,
0}, {-1, -1}}, {{-1, 0}, {-1, 0}}, {{-1, 0}, {0, -1}}, {{-1,
0}, {0, 0}}, {{-1, 0}, {1, -1}}, {{-1, 0}, {1, 0}}, {{-1,
1}, {-1, -1}}, {{-1, 1}, {-1, 0}}, {{-1, 1}, {0, -1}}, {{-1,
1}, {0, 0}}, {{0, -1}, {0, -1}}, {{0, -1}, {0, 0}}, {{0, -1}, {0,
1}}, {{0, -1}, {1, -1}}, {{0, -1}, {1, 0}}, {{0, -1}, {1, 1}}, {{0,
0}, {-1, -1}}, {{0, 0}, {-1, 0}}, {{0, 0}, {-1, 1}}, {{0,
0}, {0, -1}}, {{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}, {{0,
0}, {1, -1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0,
1}, {-1, -1}}, {{0, 1}, {-1, 0}}, {{0, 1}, {-1, 1}}, {{0,
1}, {0, -1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, -1}, {0,
0}}, {{1, -1}, {0, 1}}, {{1, -1}, {1, 0}}, {{1, -1}, {1, 1}}, {{1,
0}, {-1, 0}}, {{1, 0}, {-1, 1}}, {{1, 0}, {0, 0}}, {{1, 0}, {0,
1}}, {{1, 0}, {1, 0}}, {{1, 0}, {1, 1}}, {{1, 1}, {-1, 0}}, {{1,
1}, {-1, 1}}, {{1, 1}, {0, 0}}, {{1, 1}, {0, 1}}}

```