On the Matrix-Element Expansion of a Circulant Determinant

TL;DR

This paper derives a formula for expanding the determinant of a circulant matrix into a sum over products of matrix entries, using generating functions and combinatorial sums.

Contribution

It introduces a novel explicit formula for the coefficients in the matrix-element expansion of a circulant determinant, connecting it with restricted partition functions.

Findings

Provides a finite sum formula for expansion coefficients

Connects determinant expansion with combinatorial partition functions

Enables explicit computation of coefficients for circulant matrices

Abstract

The determinant of an $N \times N$ circulant matrix $M = {\rm CIRC}[x_0, x_1, ..., x_{N-1}$] can be expanded in the form det$ ~M= \sum 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.