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On the permanent of Schur's matrix

Published online by Cambridge University Press:  09 April 2009

R. L. Graham  and
D. H. Lehmer
R. L. Graham
Affiliation:
Bell Laboratories Murray Hill, N. J. 07974U.S.A.
D. H. Lehmer
Affiliation:
University of California, Berkeley, California, U. S. A.
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Abstract

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Schur's matrix Mn is ordinarily defined to be the n by n matrix (εjk), 0 ≦ j, k < n, where ε = exp (2 πi/n). This matrix occurs in a variety of areas including number theory, statistics, coding theory and combinatorics. In this paper, we investigate Pn, the permanent of Mn, which is define by where π ranges over all n! permutations on {0,1, …, n – 1}. Pn occurs, for example, in the study of circulants. Specifically, let Xn denote the n by n circulant matrix (xi, j) with xi, j = xi, j, where the subscript is reduced modulo n. The determinant of Xn is a homogeneous polynomial of degree n in the xi and can be written as Then Pn = A (1,1, … 1). Typical of the results established in this note are: (i) P2n = 0 for all n, (ii) Pp ≡ p ! (mod p3) for p a prime >3. (iii) If pa divides n then divides Pn. Also, a table of values of Pn is given for 1 ≦ n ≦ 23.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 4 , June 1976 , pp. 487 - 497
Copyright
Copyright © Australian Mathematical Society 1976

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