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arXiv · 2307.02652

Palindromicity of the numerator of a statistical generating function

Rebecca Bourn·William Q. Erickson
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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.

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Rebecca Bourn, William Q. Erickson. 2023-07-05. Palindromicity of the numerator of a statistical generating function. https://arxiv.org/abs/2307.02652

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