Extension of the area-bounce-exchanging bijection

Construct a bijective map on a broader class of Dyck paths that exchanges the area and bounce statistics, extending the area-bounce-exchanging bijection currently defined only on a subset of Dyck paths.

Background

The paper studies an area-bounce-exchanging bijection introduced by Ayyer and Sundaravaradan, which is defined on a subset of Dyck paths. The authors use this unresolved extension problem as one of their OpenEvolve case studies, seeding the evolutionary search with the existing partial bijection and asking the system to generalize it.

The reported experiments do not solve the problem. OpenEvolve produces a higher-scoring procedure, but it relies on a search algorithm rather than an algorithmic bijection and therefore does not provide the desired structural construction.

References

This problem is well-suited for OpenEvolve because the existing implementation serves as a starting point, and extending this bijection to a broader class of Dyck paths would be progress on an open research question.

— Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction  (2511.20987 - Brown et al., 26 Nov 2025) in Section 1, Introduction; Section 3, Area-bounce exchanging bijection

Does the deficit-zero rigidity argument have an analogue for $area$ and $bounce$, leading to a computational obstruction for an area--bounce exchange?

— A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry  (2609.05005 - Baek et al., 4 Sep 2026) in Section “Related work and open problems”, item 3

For the closely related exchange of $area$ and $bounce$, Ayyer and Sundaravaradan construct an explicit bijection on an exponentially large subset of the Dyck paths, while the full problem remains open.

— A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry  (2609.05005 - Baek et al., 4 Sep 2026) in Section “Related work and open problems”