Han–Mao–Zeng and Baril–Kirgizov descent conjectures

The conjectures concern permutation statistics on $\mathfrak{S}_n$ . First, the generating function formed from $G_n(u,v)=\sum_{\pi\in\mathfrak{S}_n}u^{\operatorname{des}_2(\pi)}v^{\operatorname{cyc}(\pi)}$ is conjectured to have the continued-fraction expansion stated by Han, Mao, and Zeng; this is their reformulation of a conjecture of Baril and Kirgizov. Second, for every $n\ge 0$ , the bistatistic $(\operatorname{des}_2,\operatorname{ear})$ is symmetric, namely $\sum_{\pi\in\mathfrak{S}_n}u^{\operatorname{des}_2(\pi)}v^{\operatorname{ear}(\pi)}=\sum_{\pi\in\mathfrak{S}_n}u^{\operatorname{ear}(\pi)}v^{\operatorname{des}_2(\pi)}$ , where $\operatorname{ear}$ is the number of exclusive antirecord cycle peaks.

Progress summary

Solved

A new unrefereed preprint says both conjectures are proved, but the claim has not yet been independently verified.

The conjectures concern equidistribution identities for refined permutation statistics, including the Han–Mao–Zeng/Vajnovszki identity and two Baril–Kirgizov bistatistics. Baril and Kirgizov formulated the latter conjectures after proving equidistribution for several individual statistics.

Known results

The 2021 paper proves $(\operatorname{des}_{2},\operatorname{des})\sim(\operatorname{pex},\operatorname{exc})$ and gives a bijective proof.
Its Corollary 1.6 establishes four equidistributed bistatistics, including $(\operatorname{exc},\operatorname{pex})$ and $(\operatorname{des},\operatorname{des}_{2})$ .
Its Theorem 1.7 supplies five equidistributed companions for the Baril–Kirgizov conjecture.

August 2026 proof claim

Taifeng Ding and Sherry H. F. Yan report that both conjectural identities are established and that the symmetry result has five equidistributed companions. The manuscript is unrefereed, so this strengthened resolution remains unconfirmed.

Current status (as of August 2026): The identities have published-preprint proof claims, while independent verification of the new strengthening remains open.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On two conjectures concerning special kinds of descents on permutations — arXiv — Taifeng Ding, Sherry H. F. Yan

algebraic-combinatorics

2 days ago