Qiu–Remmel conjecture on quadrant marked mesh patterns

For each $n\geq 0,$ the distribution of quadrant marked mesh-pattern occurrences on the set of $132$ -avoiding permutations $\operatorname{Av}_n(132)$ is the distribution conjectured by Qiu and Remmel. The supplied sources do not state the specific quadrant marked mesh patterns or the conjectured distribution explicitly.

Progress summary

Solved

A new, not-yet-reviewed preprint claims to settle the conjecture and correct a serious error in the earlier proof.

The Qiu–Remmel conjecture concerns the distribution of quadrant marked mesh patterns in $132$ -avoiding permutations. The new paper also revisits the proof context of the earlier publication.

August 2026 preprint

Shiqi Cao, Sergey Kitaev, and Yuxin Wu claim to prove the conjectured distribution for $132$ -avoiding permutations and identify a crucial error in the prior publication. This is a substantial claimed resolution, but the preprint is unreviewed and has not been independently corroborated in the retrieved record.

Current status (as of August 2026): The conjectured distribution is claimed in a new preprint, but remains unverified; the earlier proof's error is reported, and no independently corroborated resolution is recorded.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Wilf Equivalence for Length-Three Patterns and Flat POPs, and a Conjecture of Qiu and Remmel — arXiv — Cao, Shiqi, Kitaev, Sergey, Wu, Yuxin

enumerative-combinatorics permutation-patterns

21 hours ago