The Schur matrix 
of order 
is the 
square matrix with entries 
for 
, where 
is a positive integer
and 
is a primitive
root of unity (Graham and Lehmer 1976). Here 
is the imaginary unit. The
Schur matrix has a particularly simple determinant
given by
 |
(1)
|
where 
is an odd prime and
 |
(2)
|
This determinant has been used to prove the quadratic reciprocity theorem (Landau 1958, Vardi 1991). The absolute
values of the permanents of the Schur matrix of
order 
,
for 
, 1, ..., are given by 1, 3, 5, 105,
81, 6765, ... (absolute values of OEIS A003112;
Vardi 1991).
Denote the Schur matrix 
with the first row and first column omitted by 
. Then
 |
(3)
|
where perm denotes the permanent (Vardi 1991).
