Polynomials Arising from Sorted Binomial Coefficients

TL;DR

This paper studies the properties of Pascalian polynomials derived from sorted binomial coefficients, analyzing their roots, asymptotics, and algebraic structure to unify and extend previous combinatorial observations.

Contribution

It introduces Pascalian polynomials from sorted binomial coefficients, deriving recursions, root bounds, and asymptotic behavior, and explores their algebraic properties.

Findings

Roots of $P_n(z)$ converge to a specific curve in the complex plane

The roots asymptotically fill the curve densely

Discussion of reducibility and Galois groups of $P_n(z)$

Abstract

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated study. Here we refer to $\left\langle {n \atop k} \right\rangle$ as the Pascalian numbers and unify the various perspectives of $\left\langle {n \atop k} \right\rangle$. We then view each row of the $\left\langle {n \atop k} \right\rangle$ triangle as the coefficients of the $n$th Pascalian polynomial, which we denote $P_n(z)$. We derive recursions, formulae, and bounds on $P_n(z)$'s roots in $\mathbb{C}$, and characterize the asymptotics of these roots. We show the roots of $P_n(z)$ converge uniformly to a curve $\partial \Gamma \subset \mathbb{C}$ and asymptotically fill the curve densely. We conclude with a discussion of the reducibility and Galois groups of $P_n(z)$. Our work has natural connections to the truncated binomial polynomials, asymptotic analysis, and well-known integer families.