Schur Matrix
The 
Square Matrix formed by setting 
, where 
is a 
th Root of Unity.The Schur matrix has a particularly simple Determinant given by
where
is an Odd Prime and
This determinant has been used to prove the Quadratic Reciprocity Law (Landau 1958, Vardi 1991). The AbsoluteValues of the Permanents of the Schur matrix of order
are given by 1, 3, 5,105, 81, 6765, ... (Sloane's A003112, Vardi 1991).Denote the Schur matrix 
with the first row and first column omitted by 
. Then
where perm denoted the Permanent (Vardi 1991).
References
Graham, R. L. and Lehmer, D. H. ``On the Permanent of Schur's Matrix.'' J. Austral. Math. Soc. 21, 487-497, 1976. Landau, E. Elementary Number Theory. New York: Chelsea, 1958. Sloane, N. J. A. SequenceA003112/M2509in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 119-122 and 124, 1991.
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