Short Proofs in Algebraic and Enumerative Combinatorics

TL;DR

This paper provides concise proofs for several open problems in algebraic and enumerative combinatorics, including conjectures on modular lattices, parking functions, and plactic monoids, achieved autonomously by ChatGPT 5.4 Pro.

Contribution

It introduces new short proofs for multiple open problems in combinatorics, demonstrating the effectiveness of AI-generated proofs in mathematical research.

Findings

Resolved a conjecture on the echelonmotion operator on modular lattices.

Proved conjectures related to parking functions studied by Stanley, Yin, and Hopkins.

Settled two conjectures on centralizers in the plactic monoid.

Abstract

We present several short proofs that resolve open problems from the algebraic and enumerative combinatorics literature. First, we consider the echelonmotion operator on modular lattices. We resolve a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas, and Williams and, consequently, obtain a new algebraic bijective proof of a classical result of Dilworth. Second, we consider statistics on parking functions studied by Stanley and Yin and by Hopkins. We prove some conjectures of Hopkins. Third, we consider centralizers in the plactic monoid. We settle two conjectures of Sagan and Wilson. All of these proofs were obtained autonomously by ChatGPT 5.4 Pro.