Re permanents (cont)
N. J. A. Sloane
njas at research.att.com
Tue Oct 28 16:51:35 CET 2003
There were several errors in my previous message. Thanks to
Wouter for getting a(4). Here is the cortrected version
of A087981 and the new A087982. NJAS
%I A087981
%S A087981 0,2,4,24,128,880,6816,60032,589312
%N A087981 Permanent of an n X n (+1,-1)-matrix with exactly n-1 -1's on the diagonal.
%C A087981 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.
%C A087981 I don't know for which values of n this has been confirmed - compare A087982. - njas
%C A087981 The maximal 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.
%K A087981 nonn,easy,more,new
%Y A087981 Cf. A087982.
%O A087981 2,2
%A A087981 Gordon Royle (gordon(AT)csse.uwa.edu.au), Oct 28 2003
%I A087982
%S A087982 1,0,2,8,24,128,880,6816,60032,589312
%N A087982 Conjectured values for maximal permanent of a nonsingular n X n (+1,-1)-matrix.
%C A087982 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).
%C A087982 a(1) through a(4) have been proved. I don't know if any later values have been rigorously established.
%C A087982 The maximal possible value for the permanent of a singular n X n (+1,-1)-matrix is obviously n!.
%D A087892 A. R. Kraeuter and N. Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.
%D A087982 N. Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
%D A087892 Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
%e A087982 a(4) = 8 from the following matrix:
%e A087982 -1 +1 +1 +1
%e A087982 +1 +1 +1 +1
%e A087982 +1 -1 +1 -1
%e A087982 -1 +1 +1 -1
%K A087982 nonn,easy,more,new
%Y A087982 Cf. A087982.
%O A087982 2,2
%A A087982 njas, Oct 28 2003
%E A087982 a(4) >= 8 from Edwin Clark, a(4) = 8 from Wouter Meeussen.
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