## Spectrum of the determinant function

The spectrum of the determinant function for {-1,1} matrices is defined to be the set of values taken by |det Rn| / 2n-1 as Rn ranges over all n×n {-1,1} matrices. For n up to size 7, the spectrum includes all integers between 0 and the maximal determinant (divided by 2n-1). The spectrum for n=8 was first computed by Metropolis, Stein, and Wells [M1], who found that gaps occur: in particular, there are no matrices with determinant 19, 21-23, or 25-31 (times 27). A non-computer proof of the existence of gaps was later given by Craigen. At present, the spectrum is known up to size n=11 and for n=13. Conjectures have been formulated up to size n=22. In the tables below we use interval notation [a,b] to indicate the set of integers between a and b, inclusive.

Spectrum for n=1
1

Spectrum for n=2
[0,1]

Spectrum for n=3
[0,1]

Spectrum for n=4
[0,2]

Spectrum for n=5
[0,3]

Spectrum for n=6
[0,5]

Spectrum for n=7
[0,9]

Spectrum for n=8
[0, 18]
20
24
32

Spectrum for n=9
[0, 40]
42
[44, 45]
48
56

Spectrum for n=10
[0, 102]
[104, 105]
108
110
112
[116, 117]
120
125
128
144

Spectrum for n=11
[0, 268]
[270, 276]
[278, 280]
[282, 286]
288
291
[294, 297]
304
312
315
320

Conjectured spectrum for n=12
Spectrum for n=13 - recently proved complete by Brent, Orrick, Osborn, and Zimmermann
Conjectured spectrum for n=14
Conjectured spectrum for n=15
Conjectured spectrum for n=16
Conjectured spectrum for n=17

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Page created 3 August 2009.
Last modified 16 April 2010.
Comments: maxdet@indiana.edu