Exact hook-length inequality for 3-regular partitions

Prove that the total number of hooks of length 2 in all 3-regular partitions of n is at least the total number of hooks of length 1 in all 3-regular partitions of n for every integer n≥28.

Background

The paper defines b_{ℓ,t}(n) as the total number of hooks of length t in all ℓ-regular partitions of n. Singh and Barman conjectured the inequality b_{3,2}(n)≥b_{3,1}(n) for every n≥28. The present paper proves this inequality only asymptotically, via Corollary 1.4, so the stated pointwise assertion for every n≥28 remains unresolved here.

References

Singh and Barman [15] proved that b2,2(n) ≥ b2,1(n) for all n > 4 and b2,2(n) ≥ b2,3(n) for all n ≥ 0, and conjectured that b3,2(n) ≥ b3,1(n) for n ≥ 28 (see Theorems 1.4, 1.5, and Conjecture 6.1). Corollary 1.4 generalizes their theorems and proves the conjecture asymptotically.

— Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions  (2501.10916 - Kim, 19 Jan 2025) in Section 1, Introduction