Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Chen, Rong
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214083924