On the Matrix-Element Expansion of a Circulant Determinant
Abstract
The determinant of an
N×N
circulant matrix
M=CIRC[
x
0
,
x
1
,...,
x
N−1
] can be expanded in the form det
M=∑
C
a
0
a
1
...
a
N−1
x
a
0
x
a
1
...
x
a
N−1
. By using the generating function of a restricted, mod-
N
partition function, we derive a formula for the coefficients in this expansion as finite sums over products of binomial coefficients with integer variables.