Palindromicity of the numerator of a statistical generating function

TL;DR

This paper proves a conjecture about the palindromic and unimodal nature of certain polynomials related to the discrete earth mover's distance, connecting combinatorial and algebraic properties.

Contribution

It establishes the palindromicity of polynomials arising from generating functions in EMD and links their properties to Young diagram symmetries, providing explicit formulas.

Findings

Proved palindromicity and unimodality of $N_n(t)$

Connected polynomial properties to Young diagram symmetries

Derived explicit formulas for coefficients and expected EMD value

Abstract

We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials $N_n(t)$. These recursively defined polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD). The key to our proof is showing that the defining recursion can be viewed as describing sums of symmetric differences of pairs of Young diagrams; in this setting, palindromicity is equivalent to the preservation of the symmetric difference under the transposition of diagrams. We also observe a connection to recent work by Defant et al. (2024) on the Wiener index of minuscule lattices, which we reinterpret combinatorially to obtain explicit formulas for the coefficients of $N_n(t)$ and for the expected value of the discrete EMD.