A086900
Number of real n X n symmetric (0,1) matrices with positive determinant.
Original entry on oeis.org
1, 1, 5, 338, 14186, 526876, 52658844, 28076946520, 18518751047608, 13637385623943256
Offset: 1
For n = 2 the only example is the identity matrix.
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triamat[li_List] := Block[{len=Sqrt[8Length[li]+1]/2-1/2}, If[IntegerQ[len], Part[li, # ]& /@ Table[If[j>i, j(j-1)/2+i, i(i-1)/2+j], {i, len}, {j, len}], li]]; Table[it=triamat/@ IntegerDigits[Range[0, -1+2^(n(n+1)/2)], 2, n(n+1)/2]; Count[it, (q_)?MatrixQ/;(Det[q]>0)], {n, 5}]
A118993
Number of real n X n symmetric (+1,-1) matrices with nonzero permanent.
Original entry on oeis.org
2, 4, 64, 832, 23808, 1725952, 268435456, 64638447616, 33770336417792
Offset: 1
A118999
Number of real n X n symmetric (+1,-1) matrices with negative permanent.
Original entry on oeis.org
1, 0, 32, 240, 11904, 533568, 134217728, 22741568512, 16885168208896
Offset: 1
A119001
Minimal permanent of real n X n symmetric (+1,-1) matrices.
Original entry on oeis.org
-1, 0, -6, -8, -120, -96, -5040, -4320, -362880
Offset: 1
A119010
Number of symmetric n X n (+1,-1)-matrices over the reals with zero permanent.
Original entry on oeis.org
0, 4, 0, 192, 8960, 371200, 0, 4081029120, 1414035671040
Offset: 1
A118994
Number of real n X n symmetric (+1,-1) matrices with positive determinant.
Original entry on oeis.org
1, 0, 16, 432, 8448, 282240, 81949952, 32715189248, 12792558313472, 9318420858593280
Offset: 1
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F:= proc(n) local Q,q,X,x,t,A,ii,L,v;
Q:= [[1,1],seq(seq([i,j],i=2..j),j=2..n)];
q:= nops(Q);
X:= [seq(x[q[1],q[2]],q=Q)];
t:= 0:
A:= Matrix(n,n,shape=symmetric,symbol=x);
A[2..n,1]:= Vector(n-1,1);
for ii from 0 to 2^q-1 do
L:= map(s -> 2*s-1, convert(2^q+ii,base,2)[1..q]);
v:= LinearAlgebra:-Determinant(subs(zip(`=`,X,L),A));
if v > 0 then t:= t+1 fi
od;
2^(n-1)*t;
end proc:
seq(F(n),n=1..7); # Robert Israel, Apr 14 2016
A105641
Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1).
Original entry on oeis.org
0, 1, 2, 5, 14, 39, 111, 322, 947, 2818, 8470, 25677, 78420, 241061, 745265, 2315794, 7228702, 22656505, 71273364, 224965675, 712249471, 2261326010, 7197988973, 22966210236, 73437955105, 235307698544, 755395560220, 2429293941019
Offset: 2
a(4)=2 because we have UUDUDUDD and UUUDUDDD.
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G:=((1+z)^2-sqrt((1+z^2)^2-4*z))/2/z/(2+z+z^2)-1: Gser:=series(G,z=0,36): seq(coeff(Gser,z^n),n=2..32);
Showing 1-7 of 7 results.
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