Hook-length bias inequality for even indices

Prove that for every even integer k≥8 and every n≥0 with n≠k+1, the total number b_{2,k}(n) of hooks of length k in all 2-regular partitions of n is at least the total number b_{2,k+1}(n) of hooks of length k+1 in all 2-regular partitions of n.

Background

The paper studies inequalities between b_{t,i}(n), the total numbers of hooks of a specified length i in all t-regular partitions of n. Singh and Barman conjectured that b_{2,k}(n)≥b_{2,k+1}(n) for every k≥3 and all n≥0 except n=k+1. The paper proves that this conjecture fails for every odd k≥3, while establishing it for k=4 and k=6.

Based on computational verification for 8≤k≤20 and n≤10000, the authors formulate a modified conjecture restricted to even k≥8. This remains an unresolved extension of the verified cases k=4 and k=6.

References

After checking $8\le k\le 20$ and $n\le 10000$, we propose the following conjecture, which is a modification of Conjecture (ii). For even $k\ge 8$, $b_{2,k}(n)\ge b_{2,k+1}(n)$ for all $n\ge 0$ and $n\ne k+1$.

— On the hook length biases of the $2$- and $3$-regular partitions  (2501.13753 - Qu et al., 23 Jan 2025) in Section 1, Introduction, immediately after Conjecture 2