Palindromicity of the numerator of a statistical generating function
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909842373869568 |
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| author | Bourn, Rebecca Erickson, William Q. |
| author_facet | Bourn, Rebecca Erickson, William Q. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02652 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Palindromicity of the numerator of a statistical generating function Bourn, Rebecca Erickson, William Q. Combinatorics 05A15 (Primary) 11B37, 05A17, 05C09 (Secondary) 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. |
| title | Palindromicity of the numerator of a statistical generating function |
| topic | Combinatorics 05A15 (Primary) 11B37, 05A17, 05C09 (Secondary) |
| url | https://arxiv.org/abs/2307.02652 |