e-regular and e-restricted partitions

Let $e\in \{2,3,4,\dots \}\cup \{\mathrm{\infty }\}$ and let $\lambda ⊢n$ be a partition with parts ${\lambda }_1\ge {\lambda }_2\ge \cdots$ and conjugate ${\lambda }^{\mathrm{\prime }}$ (Partitions, English diagrams, and conjugation). Then $\lambda$ is $e$-regular when every positive integer occurs as a part of $\lambda$ fewer than $e$ times, i.e. $z_j(\lambda )<e$ for every $j\ge 1$, and $e$-restricted when ${\lambda }_i-{\lambda }_{i+1}<e$ for every $i\ge 1$ (with ${\lambda }_i:=0$ for $i>\mathrm{\ell }(\lambda )$).

For prime $e=p$ these are exactly the $p$-regular and $p$-restricted notions of p-regular and p-restricted partitions; the same inequalities define the extension to composite finite $e$. For $e=\mathrm{\infty }$ both conditions are vacuous, so every partition is both $e$-regular and $e$-restricted: the inequalities $z_j(\lambda )<\mathrm{\infty }$ and ${\lambda }_i-{\lambda }_{i+1}<\mathrm{\infty }$ hold because partitions have finitely many nonzero parts and each difference is a nonnegative integer. Moreover $\lambda$ is $e$-restricted if and only if its conjugate ${\lambda }^{\mathrm{\prime }}$ is $e$-regular: the multiplicity of the part $i$ in ${\lambda }^{\mathrm{\prime }}$ is the number of columns of $[\lambda ]$ of height exactly $i$, namely ${\lambda }_i-{\lambda }_{i+1}$, so the two conditions $z_i({\lambda }^{\mathrm{\prime }})={\lambda }_i-{\lambda }_{i+1}<e$ are the same family of inequalities, and for $e=\mathrm{\infty }$ both sides hold. In particular $\mathrm{\varnothing }$ is simultaneously $e$-regular and $e$-restricted for every $e$ (p-regular and p-restricted partitions).