## 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 R_{n}| / 2^{n-1} as
R_{n} 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 2^{n-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 2^{7}). 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