Polynomials Arising from Sorted Binomial Coefficients

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.