## 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 Absolute Values 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. Sequence A003112/M2509 in 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.