Real-rootedness of the type A minuscule polynomials

TL;DR

This paper proves conjectures about the real-rootedness and properties of certain polynomials related to earth mover's distance and minuscule lattices, revealing their asymptotic normality and total positivity.

Contribution

It establishes the real-rootedness of the polynomials $N_n(t)$ and explores their coefficient properties, including asymptotic normality and total positivity.

Findings

Proved the real-rootedness of $N_n(t)$ polynomials.

Showed the coefficients of $N_n(x)$ are asymptotically normal.

Demonstrated the total positivity and $x$-log-concavity of the polynomial sequence.

Abstract

We prove two recent conjectures of Bourn and Erickson (2023) regarding the real-rootedness of a certain family of polynomials $N_n(t)$ as well as the sum of their coefficients. These polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD) and have also connection to the Wiener index of minuscule lattices. We also prove that the coefficients of $N_n(x)$ are asymptotically normal, the coefficient matrix of $N_n(x)$ is totally positive and the polynomial sequence $N_n(x)$'s is $x$-log-concave.