Maximizing the adjacent comparison classes of parking functions

We study the sizes of adjacent-comparison classes of parking functions, as posed in Problem 54 of Ari Cruz et al., On Some Discrete Statistics of Parking Functions (Journal of Integer Sequences 27 (2024), Article 24.8.6). The letters A, D and T encode strict ascents, strict descents and equalities. The manuscript gives explicit injections, inverse maps and strictness witnesses that classify all maximizing encodings. For odd n >= 3, only the alternating encoding beginning with A is maximal. For even n >= 4, both alternating encodings are maximal. For n = 2, A, D and T each have one parking function; for n = 1, the empty encoding has one. An exact subset recurrence counts the maximum class. The alternating counts themselves predate this manuscript (OEIS A260694 and A264963). This record contains a seven-page English preprint PDF, the complete GitHub v1.0.0 reproducibility archive (including LaTeX source, a full Chinese proof, Python and C++ code, exact data and audit results), and a deposit README. The PDF and archive are byte-identical to the public GitHub snapshot at commit 57744603e135bf7a2ea215bc9884490e900edeca. The first public release was on October 28, 2026. The supplied checks compare three independent exact counting methods through n = 6, verify the composed injection and inverse on all 18,248 parking functions of lengths at most six, compare all ternary encoding classes through n = 10, and check maximum values through n = 20. Finite computation audits implementations; the universal mathematical arguments are given in the manuscript. OpenAI Codex assisted extensively with mathematical exploration, proof development and checking, enumeration, programming, literature screening, English exposition and publication preparation. This is an AI-assisted research preprint. Independent expert peer review, proof-assistant formalisation and certified historical priority are not claimed. Licenses: the manuscript (PDF and LaTeX), documentation and JSON data are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Python, C++ and verification automation are licensed under MIT. These material-specific scopes are detailed in LICENSES.md inside the archive; the two licenses apply to their respective file categories. Code and living repository: https://github.com/KevinMaSAI/parking-function-encoding-maximaAuthor: Qiyuan Ma, Shanghai Jiao Tong University. ORCID: 0009-0000-9532-0821.