Palindromicity of the numerator of a statistical generating function

TL;DR

This work proves the BW conjecture that the numerators $N_n(t)$ of the generating functions for the one-dimensional earth mover's distance are palindromic and unimodal. By reframing the recursion as sums of symmetric differences of pairs of Young diagrams and exploiting transposition symmetry, the authors establish palindromicity; they further connect to Defant et al.'s Wiener index for minuscule lattices of Type A to obtain explicit coefficient formulas for $N_n(t)$ and an exact expression for the expected EMD. The explicit Wiener-index formulas yield both the coefficient structure of $N_n(t)$ and a closed form for $ ext{E}[ ext{EMD}]$, enabling unimodality to follow and enabling closure of BW’s recursion. Additional observations include $N_n(1) = S(n-1)$ with $S(n)=n2^{2n-1}$ and the real-rootedness of $N_n(t)$, plus a suggested Gorenstein interpretation via Hilbert series of a potential ring.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 3.1
• proof
• Corollary 3.2