An involution on Dyck paths in the region defc≤min(area,dinv) that interchanges area and dinv

@misc{hawkes2026involution,
  title = {{An involution on Dyck paths in the region $\defc\le\min(\area,\dinv)$ that interchanges area and dinv}},
  author = {Graham Hawkes},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.21403},
  eprint = {2609.21403},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21403},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - An involution on Dyck paths in the region defc≤min(area,dinv) that interchanges area and dinv
AU  - Graham Hawkes
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.21403
UR  - https://arxiv.org/abs/2609.21403
AB  - We construct an explicit involution on Dyck paths satisfying defc≤min(area,dinv) that interchanges area and dinv, where defc=binom n2−area−dinv and n is the semilength. In addition, we give an involution on Dyck paths with defc ≤ 2n−8 that interchanges area and dinv. Finally, we give an explicit partition formula for the portion of the q,t-Catalan polynomial of total degree at least binom n2−2n+8, thereby proving a conjecture of Lee and Li~\cite[Conjecture~4]{LeeLi11}. Altogether our results give a combinatorial explanation of q,t-Catalan symmetry in the region defc≤max(2n−8,min(area,dinv)). Our constructions rely on a number of new combinatorial objects developed here and three main external tools: the dual Dyck insertion of \cite{Hawkes26}, the Garsia--Milne involution principle~\cite{GarsiaMilne81},~\cite{Doyle19}, and the partition bijection of ~\cite{LoehrWarrington09}.
ER  -