Berkovich–Dhar hook-length conjectures

Let $\mathcal{O}_{n,m}$ be the set of partitions $\lambda$ of $n$ whose parts are odd and whose largest part satisfies $\lambda_1\le m$ , and let $\mathcal{D}_{n,m}$ be the set of partitions of $n$ into distinct parts with $\lambda_1\le m$ . If $h_2(\lambda)$ denotes the number of cells of the Ferrers diagram of $\lambda$ having hook length $2$ , then the hook-pair conjecture asserts that, for all $n,m\ge 0$ , $\displaystyle \sum_{\lambda\in\mathcal{O}_{n,m}}\binom{h_2(\lambda)}{2}\ge \sum_{\lambda\in\mathcal{D}_{n,m}}\binom{h_2(\lambda)}{2}$ . Berkovich and Dhar also formulated a stronger coefficientwise inequality for the corresponding hook-length generating functions; the supplied source does not state that inequality explicitly.

References

Primary source

arXiv

Additional references

Proofs of two conjectures of Berkovich and Dhar on hook lengths — arXiv — Rong Chen

combinatorics partition-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle both conjectures and strengthen them, but the result has not yet been independently checked.

The Berkovich–Dhar conjectures compare hook-length statistics for odd and distinct partitions. They were posed by Berkovich and Dhar; the latest claim establishes coefficientwise domination through weight-preserving injections and product identities.

Known results

Berkovich and Dhar proved the finite unweighted inequality, while retaining the weighted inequality as Conjecture 2.6 and the unbounded statement as Conjecture 4.1.
Ballantine, Burson, Craig, Folsom, and Wen (2023) proved the unweighted bias for hook lengths $2$ and $3$ and conjectured the analogue for every fixed hook length.
Craig, Dawsey, and Han proved the unbounded unweighted inequality for all sufficiently large $n$ , via asymptotics for every hook length $h\geq 2$ .

September 22, 2026 claimed resolution

Rong Chen's preprint, Proofs of two conjectures of Berkovich and Dhar on hook lengths, claims proofs of both named conjectures in stronger coefficientwise form. The announcement is unconfirmed because the preprint is unrefereed.

Current status (as of September 2026): the earlier unweighted results are established, while Chen's claimed resolution of the two weighted conjectures remains unverified.

Sources

No solutions have been posted yet.