Dai–Fu–Qiu open problem on rational Dyck paths

Let $\mathcal{D}_{m,n}$ denote the set of rational Dyck paths with parameters $m,n\in\mathbb{Z}_{>0}$ . For the parameter range $m<n$ , find suitable run-type and return-type statistics $R$ and $T$ on $\mathcal{D}_{m,n}$ , extending the statistics studied by Dai, Fu, and Qiu, such that their joint distribution is symmetric; in particular, establish an identity of the form $\#\{P\in\mathcal{D}_{m,n}:R(P)=i,\ T(P)=j\}=\#\{P\in\mathcal{D}_{m,n}:R(P)=j,\ T(P)=i\}$ for all admissible $i,j$ , together with an explicit enumeration of paths having prescribed statistic values. The supplied sources do not specify the exact modified statistics or the precise convention for $\mathcal{D}_{m,n}$ .

References

Primary source

arXiv

Additional references

Symmetries on vector parking functions via bounded lattice paths — arXiv — Wenjie Fang, Yang Li, Zhicong Lin

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new preprint claims to settle the problem, but no independent confirmation has appeared.

Dai, Fu, and Qiu posed the problem in their 2025 study of refined statistics on rational Dyck paths. It asks for suitable statistics with symmetric joint distributions beyond the range covered by their initial involutions.

Known results

Dai, Fu, and Qiu (2025) constructed involutions exchanging run-type statistics with return for $m \ge n$ .
Their paper left the modification for $m<n$ open and noted that explicit enumeration with prescribed run and return counts also remained open.

October 2026 claimed resolution

Fang, Li, and Lin claim new composition-run/return symmetries for bounded lattice paths, transferred to vector parking functions, and state that these resolve the cited rational-Dyck-path problem. The claim is supported by a specialist preprint but has not received independent mathematical assessment in the retrieved sources.

Current status (as of October 2026): A specialist preprint claims to resolve the Dai–Fu–Qiu problem, but that resolution is unverified; the broader vector-parking-function program remains incomplete.

Sources

No solutions have been posted yet.