Re Permanents of (+1,-1) matrices

[Edwin Clark]

On Tue, 28 Oct 2003, N. J. A. Sloane wrote:

> 
> Thanks to everyone who replied.  To summarize, let A
> be a nonsingular +1,-1  n X n matrix.  The conjecture appears
> to be that for n >= 5 the max permanent of A is given by
> the following (new) sequence:
> 
> %I A087981
> %S A087981 0,2,4,24,128,880,6816,60032,589312
> %N A087981 Permanent of the n X n (+1,-1)-matrix with exactly n-1 -1s on the diagonal.
> %C A087981 It is conjectured by Kraeuter and Seifter that for n >= 5 this is the maximum possible value for the permanent of a nonsingular n X n (+1,-1)-matrix with exactly n-1 -1's on the diagonal. 
> %C A087981 I don't know for what values of n this has been confirmed. As Edwin Clark points out, it is certainly false for n = 4. - njas
> %C A087981 The maximum possible value for the permanent of a singular n X n (+1,-1)-matrix is obviously n!.
> %D A087891 A. R. Kraeuter and N. Seifter, Some properties of the permanent of (1,-1)-matrices,  Linear and Multilinear Algebra 15 (1984), 207-223.
> %D A087981 N. Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
> %D A087891 Edward Tzu-Hsia  Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
> %O A087981 2,2
> %K A087981 nonn,easy,more
> %A A087981 Gordon Royle (gordon(AT)csse.uwa.edu.au), Oct 28 2003
> 
> But what is the max permanent for n <= 4 ?  For 
> n =  1  2  3  4    we have
>      1  0  6  x     where thanks to Ed Clark we know x >= 8
> 
> It remains to see if 8 is the max permanent of a 4x4 nonsingular
> +1,-1 matrix - could someone please do that computation?
> 
> 
> NJAS
> 

I already checked that 8 is maximum permanent of a 4x4 nonsingular
+1,-1 matrix. Also 24 is the maximum for the 5x5 case PROVIDED there is a
row of all 1's. Perhaps someone can deduce the true maximum of the 5x5
case from this fact?

--Edwin