Burstein’s conjecture on descent-top sets

For every $n\ge 1$ and every subset $S\subseteq [n]=\{1,\ldots,n\}$ , the numbers of permutations in the two avoidance classes having descent-top set $S$ are equal:

\#\{\pi\in\operatorname{Av}_n(2143):\operatorname{DesTop}(\pi)=S\}=\#\{\pi\in\operatorname{Av}_n(3421):\operatorname{DesTop}(\pi)=S\}.

Here $\operatorname{Av}_n(\sigma)$ denotes the set of permutations of $[n]$ avoiding the pattern $\sigma$ , and $\operatorname{DesTop}(\pi)=\{\pi_i:1\le i<n,\ \pi_i>\pi_{i+1}\}$ is the set of values occurring at the tops of descents.

References

Primary source

arXiv

Additional references

Descent Tops and Pinnacles on 2143- and 3421-Avoiding Permutations — arXiv — Yue Dong, Lily Li Liu, Tongyuan Zhao

enumerative-combinatorics permutation-patterns

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A 2026 preprint claims to settle the conjecture with an explicit reversible matching between the two permutation classes, but the result has not been independently checked.

Burstein’s conjecture asserts equality of descent-top distributions for permutations avoiding $2143$ and $3421$ . A new preprint by Yue Dong, Lily Li Liu, and Tongyuan Zhao claims a stronger bijective result, also controlling pinnacle data.

Known results

Burstein’s revised paper, accepted January 7, 2025, established related descent-top equidistributions for shorter patterns and recorded analogous length- $4$ conjectures.
Zhou, Zang, and Yan reportedly proved those related length- $4$ conjectures in 2024.

October 2026 claimed bijective proof

The preprint constructs an explicit bijection $\Phi:\operatorname{Av}_n(2143)\to\operatorname{Av}_n(3421)$ preserving descent-top sets and reversing pinnacle words. It therefore claims the conjecture and a refined Dumont-permutation consequence, but the preprint is unrefereed and no independent verification was found.

Current status (as of October 2026): A preprint claims the conjecture is solved by an explicit bijection, while independent verification of that proof is not recorded.

Sources

No solutions have been posted yet.