Analytic Properties of Necklace Polynomials

Abstract

The necklace polynomials \[ M_n(x)=\frac1n\sum_{d\mid n}\mu(d)x^{n/d} \] play a central role in discrete mathematics: they count aperiodic necklaces, enumerate monic irreducible polynomials over finite fields, and give the dimensions of homogeneous components of free Lie algebras. Despite their inherently discrete origins, we show that treating $M_n(x)$ as a function of a real variable $x$ unlocks surprising structural properties that answer natural enumerative questions. In this paper, we study $M_n(x)$ as a real-variable function and establish several new analytical and monotonicity properties. We prove that the normalized functions $M_n(x)/x^n$ and their higher normalized derivatives are strictly increasing on $[1,\infty)$. As a consequence, we show that the proportion of irreducible polynomials of fixed degree over $\mathbf F_q$ increases with $q$. We also establish strict growth with respect to the degree $n$ for $x\ge2$. In addition, we determine a sharp threshold for log-convexity: the sequence $\{M_n(x)\}_{n\ge2}$ is uniformly log-convex if and only if $x>8$. These results reveal unexpected analytic structure underlying necklace polynomials and show how real-variable methods can yield new information about discrete enumeration problems. For instance, it is shown that adding one more bead to a sufficiently long necklace will approximately increase the total number of primitive, rotationally distinct configurations by a factor of the number of available colors.