PreprintCC BY 4.0arXiv
Proofs of two conjectures of Berkovich and Dhar on hook lengths
Abstract
We prove stronger forms of two conjectures of Berkovich and Dhar concerning a coefficientwise inequality and pairs of hooks of length two in bounded partitions. The coefficientwise inequality implies the corresponding inequality for hook pairs. In particular, for each fixed size and common bound on the largest part, the total number of such pairs in odd partitions is at least that in distinct partitions. We give combinatorial proofs by explicit weight-preserving injections based on Glaisher's bijection and analytic proofs using finite product identities and known hook generating functions.
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BibTeX
@misc{chen2026proofs,
title = {{Proofs of two conjectures of Berkovich and Dhar on hook lengths}},
author = {Rong Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26324},
eprint = {2609.26324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26324},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Proofs of two conjectures of Berkovich and Dhar on hook lengths
AU - Rong Chen
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26324
UR - https://arxiv.org/abs/2609.26324
AB - We prove stronger forms of two conjectures of Berkovich and Dhar concerning a coefficientwise inequality and pairs of hooks of length two in bounded partitions. The coefficientwise inequality implies the corresponding inequality for hook pairs. In particular, for each fixed size and common bound on the largest part, the total number of such pairs in odd partitions is at least that in distinct partitions. We give combinatorial proofs by explicit weight-preserving injections based on Glaisher's bijection and analytic proofs using finite product identities and known hook generating functions.
ER -