Permanents and determinants of (0,1) matrices
N. J. A. Sloane
njas at research.att.com
Tue Nov 4 01:24:33 CET 2003
Hugo asked which of those sequences should be submitted. Well, let's see:
Number of different values that can be assumed by the
determinant of a real (0,1)-matrix of order n.
Starting at n=1:
2,3,5,7,11,19,43, next term 89<=a(8)<113 (hard!)
Maximal permanent of a real non-singular (0,1)-matrix of order n:
1 1 3 11 53
Number of real non-singular (0,1)-matrices having maximal permanent=
a(from previous sequence)
1 6 18 96 600
Number of different values that can be assumed by the
permanent of a real non-singular (0,1)-matrix of order n:
1 1 3 9 31 (will Jaap Spies compute a(6)?)
Number of different values that can be assumed by the
permanent of a real singular (0,1)-matrix of order n:
1 2 4 10 32
Smallest positive integer not the permanent of a nXn (0-1) matrix.
2 3 5 13 27 119 (last term from Gordon Royle's list)
DEFINITELY YES to all the above. Hugo, would you please send them in?
(with coauthors as appropriate).
as to these:
How many times does the determinant (D) or permanent (P) of an m*m
real (0,1)-matrix assume the value n (starting at n=0).
For determinants negative values are assumed with the same counts.
m=2
D: 10,3
P: 9,6,1
m=3
D: 338,84,3
P: 265,150,69,18,9,0,1,
m=4
D: 42976,10020,1200,60
P: 27713,13032,10800,4992,4254,1440,1536,576,648,24,288,96,48,0,72,
0,0,0,16,0,0,0,0,0,1
m=5
D: 21040112,4851360,1213920,144720,43560,3600
P: 10363361,3513720,4339440,2626800,3015450,1451400,1872800,962400,...
m=6
D: 39882864736,9240051240,3868663680,768723480,418703040,63612360,
46569600,6438600,5014800,529200
P: 13906734081 2722682160 4513642920 3177532800 4466769300 2396826720
3710999520 2065521600 3253760550 1468314000 2641593600 1350475200
2210277600 1034061120 1980428400 829699200 1507653000 769932000
1405153800 535852800 1181471400
THEY SHOULD be prefixed by the case m=0:
D: 1,1
P: 1,1
AND THEN I think they should be made into two sequences.
The first being something like this:
Triangle T(n,k) read by rows, where T(n,k) = number
of times a real n X n (0,1)-matrix takes the value k,
for n >= 1, 0 <= k <= A??????(n).
with keyword tabf
Similarly for the permanent one.
Thanks!
NJAS
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