On the permanent of Schur's matrix

Abstract

AbstractSchur's matrix Mnis ordinarily defined to be thenbynmatrix (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 investigatePn, thepermanentofMn, which is define bywhere π ranges over alln! permutations on {0,1, …,n– 1}.Pnoccurs, for example, in the study of circulants. Specifically, letXndenote thenbyncirculant matrix(xi, j)withxi, j= xi, j, where the subscript is reduced modulon. The determinant ofXnis a homogeneous polynomial of degreenin thexiand can be written asThenPn= A(1,1, … 1). Typical of the results established in this note are: (i)P2n= 0 for alln, (ii)Pp≡p! (modp3)forpa prime >3. (iii) IfpadividesnthendividesPn. Also, a table of values ofPnis given for 1 ≦n≦ 23.