2 problems

Lepowsky–Zhu's Empirical Hypothesis. For $j\geq 0$ and $1\leq i\leq k$, one has

Let $r\geq 2$ and $1\leq i\leq r$. For a partition $\Lambda=(\lambda_1,\ldots,\lambda_m)$, define its $(i,\ell)$-new parts recursively by … where $\lambda_j=0$ for $j\leq 0$, and i…