OFFSET

1,3

COMMENTS

It is conjectured by Kraeuter and Seifter that for n >= 5 the maximal permanent of a nonsingular n X n (+1,-1)-matrix is attained by a matrix with exactly n-1 -1's on the diagonal (compare A087981).

This has been proved by Budrevich and Guterman. - Sergei Shteiner, Jan 21 2020

The maximal possible value for the permanent of a singular n X n (+1,-1)-matrix is obviously n!.

LINKS

Mikhail V. Budrevich, Alexander E. Guterman, Kräuter conjecture on permanents is true, arXiv:1810.04439 [math.CO], 2018.

Arnold R. Kräuter and Norbert Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.

Norbert Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.

Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.

FORMULA

a(n) = A087981(n-1) for n >= 5. - Sergei Shteiner, Jan 20 2020

EXAMPLE

a(4) = 8 from the following matrix:

-1 +1 +1 +1

+1 +1 +1 +1

+1 -1 +1 -1

-1 +1 +1 -1

CROSSREFS

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 28 2003

EXTENSIONS

a(4) = 8 from W. Edwin Clark and Wouter Meeussen, a(5) = 24 and a(6) = 128 from Jaap Spies, Oct 29 2003

STATUS

approved