All-n hook-length comparison for 3-regular partitions

Prove that the total number of hooks of length 2 among 3-regular partitions of n is at least the total number of hooks of length 1 among 3-regular partitions of n for every n greater than or equal to 28.

Background

A 3-regular partition is a partition with no part divisible by 3. The quantities b_{3,1}(n) and b_{3,2}(n) count, respectively, hooks of lengths 1 and 2 across all 3-regular partitions of n.

The paper records a conjecture of Singh and Barman asserting the inequality for every n at least 28. Corollary 1.4 establishes the comparison only for sufficiently large n, explicitly proving the conjecture asymptotically rather than settling the finite interval beginning at n=28.

References

Singh and Barman [15] proved that b_{2,2}(n) ≥ b_{2,1}(n) for all n > 4 and b_{2,2}(n) ≥ b_{2,3}(n) for all n ≥ 0, and conjectured that b_{3,2}(n) ≥ b_{3,1}(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, paragraph following Corollary 1.5; cited as Conjecture 6.1 of [15]