Uniform area–dinv-swapping bijection

Construct an explicit, uniform, semilength-preserving bijection on all Dyck paths that swaps the area and dinv statistics, namely a bijection F satisfying area(F(P))=dinv(P) and dinv(F(P))=area(P) for every Dyck path P.

Background

The q,t-Catalan polynomial has a symmetric formulation involving area and dinv, and Haglund’s zeta map provides a bijective explanation for one direction of the area–dinv correspondence. A direct bijection that simultaneously exchanges both statistics would give an object-by-object explanation of the full symmetry C_n(q,t)=C_n(t,q).

The paper proves that no map in the weighted-rank polyregular class WRP can realize such an exchange, but explicitly emphasizes that this does not prove nonexistence of a bijection in general. Thus the problem remains open beyond the computational mechanism studied in the paper.

References

Finding such a bijection is a long-standing open problem.

— 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 1, Introduction; Section 1, paragraph “Bijective proofs as computations”; Section 2, Section “Related work and open problems”