Gessel and Zhuang's shuffle-compatibility conjecture for
For a permutation , let , , and denote the permutation statistics of the length of the longest up-down run, the number of peaks, and the number of descents, respectively. A statistic is shuffle-compatible if, for any two permutations with disjoint ground sets, the multiset of its values over their shuffles depends only on the statistic values of the two permutations. Gessel and Zhuang's conjecture. The triple is shuffle-compatible. This conjecture concerns whether the joint distribution of these three permutation statistics is preserved under shuffling in the required sense. The paper states that it proves this conjecture, so the conjecture is resolved.
References
Primary source
Lihong Yang and Sherry H. F. Yan, “On a conjecture concerning the shuffle-compatible permutation statistics”, arXiv:2201.06784 (2022).
Progress summary
Yang and Yan published a proof in 2022 that claims to settle the conjecture, but its correctness is recorded here as unverified.
Gessel and Zhuang conjectured that the joint statistic is shuffle-compatible. The sources identify Yang and Yan as proving this conjecture by a bijective argument.
Known results
Baker–Jarvis and Sagan previously proved shuffle-compatibility of , but their bijection did not preserve .
July 1, 2022 publication
On July 1, 2022, Yang and Yan published On a Conjecture Concerning Shuffle-Compatible Permutation Statistics, claiming a -preserving bijection and hence a proof of the conjecture. No retrieved source reports a counterexample, gap, withdrawal, or retraction.
Current status (as of September 2026): The conjecture has a published claimed proof by Yang and Yan, but this crawl does not independently verify it.
Sources
- arxiv.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- combinatorics.org
- ui.adsabs.harvard.edu
- arxiv.org
- users.math.msu.edu
- deepmind.google
- quantamagazine.org
- deepmind.google
- quantamagazine.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
Solutions 0
No solutions have been posted yet.