Log-concavity of Reduced Composition Polynomials via Adjacent B-spline Refinement Columns

Abstract

Ardila and Doker associated a reduced composition polynomial f_c(q) with every positive integer composition c and asked whether its coefficient sequence is unimodal or, more strongly, log-concave. We prove that the coefficient sequence is log-concave for every positive integer composition, and hence is unimodal. The proof identifies the coefficients with a discrete B-spline refinement column, realizes the two endpoint deletions as adjacent columns of a common lower-degree refinement matrix, and converts their nonnegative 2-by-2 minors into the required coefficient inequalities through an exact determinant identity. Status: Public Beta; internally verified candidate proof; external mathematical review pending.