Let $k, m\in\mathbb{N}$ be two integers with $m\geq k$ (agree that $0\notin\mathbb{N}$). Consider the set
$$A_{m|k}^\neq:=\{(i_1, \ldots, i_k) \in \mathbb{N}^{\times k} \mid \ i_1,\ldots,i_k \ \text{ pairwise distinct } \ \text{ s.t. } \ \ i_1+\ldots+i_k=m\}$$
of $k$-tuples with pairwise distinct integer entries whose $\ell_1$-norm is $m$.
Question: Do you know a formula for the cardinality of $a_{m|k}$ of $A_{m|k}^\neq$ and whether these numbers have a name?
Remark: The cardinality of the superset $A_{m|k}:=\{(i_1,\ldots,i_k)\in\mathbb{N}\mid i_1+\ldots + i_k = m\}$ is $\binom{m-1}{k-1}$, and I'm interested in estimates concerning the quotients $q_{m|k}:=\frac{|A_{m|k}^\neq|}{|A_{m|k}|}$ as $m\rightarrow\infty$.