# Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction

arXiv: https://arxiv.org/abs/2511.20987 · HTML: https://mlsanity.com/paper/2511.20987/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 24 references that are in the map.
- First posted on arXiv 26 Nov 2025.

## Builds on (most cited first, 4 of 4)

- [Illuminating search spaces by mapping elites](https://mlsanity.com/paper/1504.04909/) (2015)
- [ShinkaEvolve: Towards Open-Ended And Sample-Efficient Program Evolution](https://mlsanity.com/paper/2509.19349/) (2025)
- [Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics](https://mlsanity.com/paper/2410.21272/) (2024)
- [LLM4AD: A Platform for Algorithm Design with Large Language Model](https://mlsanity.com/paper/2412.17287/) (2024)

## Built on by (most cited first, 5 of 5)

- [LLM-Guided Prompt Evolution for Password Guessing](https://mlsanity.com/paper/2604.12601/) (2026)
- [Mapping Uncharted Symmetries: Machine Discovery in Combinatorics](https://mlsanity.com/paper/2605.19063/) (2026)
- [PermuFormer: Multi-Task Pretraining for Permutation Representation in Algebraic Combinatorics](https://mlsanity.com/paper/2609.25438/) (2026)
- [Algorithmic algorithm development with LLMs: A Case Study on LLM-Usage for Contraction Order Optimization in Tensor Networks](https://mlsanity.com/paper/2606.01975/) (2026)
- [AI Discovery in Combinatorics: From Problems to Solutions and Back to Bigger Problems](https://mlsanity.com/paper/2610.16555/) (2026)

## Abstract (arXiv)

Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems utilize teams of large language models (LLMs) to generate candidate solutions to a problem as human readable code. These candidate solutions are then 'evolved' with the goal of improving them beyond what an LLM can produce in a single shot. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop. We describe some lessons learned for others in the field interested in exploring the use of these systems.