The Matrix of Chromatic Joins

Matrices M(n), one for each positive integer n, arise in the theory of Birkhoff-Lewis equations for chromatic polynomials. Students of those equations think it would be helpful to have a formula for the determinant of M(n). Finding that determinant is the main object of this paper. It has been conjectured that the determinant is a product of a power of the colour-variable λ and powers of certain polynomials in λ, those called "Beraha polynomials" by combinatorialists. That would explain the observed occurrences of Beraha polynomials in the solutions of the Birkhoff-Lewis equations for small values of n. The formula obtained in this paper verifies the conjecture. This paper is closely connected with work done by R.Dahab, D. H. Younger, and the present author on partial solutions of the Birkhoff- Lewis equations. The first two computed det M(n) up to n = 6, and so made the above-mentioned conjecture seem plausible.