2,2

Permanent of an n X n (+1,-1)-matrix with exactly n-1 -1's on the diagonal.

It is conjectured by Kraeuter and Seifter that for n >= 5 this is the maximal possible value for the permanent of a nonsingular n X n (+1,-1)-matrix. I do not know for which values of n this has been confirmed - compare A087982. - N. J. A. Sloane (njas(AT)research.att.com).

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

Degree of the "hyperdeterminant" of a multilinear polynomial on (\P^1(\C))^n, or equivalently of an element of (\C^2)^{\otimes n}: see Gelfand, Kapranov and Zelevinsky. - Eric Rains, Mar 15 2004.

Polynomials in A010027 evaluated at -1. - Ralf Stephan, Dec 15 2004

Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 1994, Corollary (14.)2.10.

A. R. Kraeuter and N. Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.

N. 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.

T. D. Noe, Table of n, a(n) for n=2..101

A. R. Kr\"auter, Permanenten - Ein kurzer \"Uberblick

Index entries for sequences related to binary matrices

Krauter and Seifter prove that the permanent of an n X n {-1, 1} matrix is divisible by 2^{n - [log_2(n)] - 1}.

Let c(n) be the permanent of the {-1, +1}-matrix of order n X n with n diagonal -1's only. Let a(n) be the permanent of the {-1, +1}-matrix of order n X n with n-1 diagonal -1's only. Then by expanding along the first row (like determinant, but with no sign) we get c(n) = -c(n-1) + (n-1) a(n-1), a(n) = c(n-1) + (n-1) a(n-1), with c(2) = 2, a(2) = 0. {c(n)} has e.g.f. exp(-2x)/(1-x), see A000023. Also a(n) = c(n) + 2*c(n-1).

Cf. A087982, A087983, A000023.

Sequence in context: A068506 A119036 A164313 this_sequence A002875 A110491 A019010

Adjacent sequences: A087978 A087979 A087980 this_sequence A087982 A087983 A087984

nonn,easy,nice

Gordon Royle (gordon(AT)maths.uwa.edu.au), Oct 28 2003

More terms from Jaap Spies (j.spies(AT)hccnet.nl), Oct 28 2003

Further terms from Gordon Royle, Oct 29 2003

Definition via e.g.f. from Eric Rains, Mar 15 2004