Let $\mathcal{C}$ be a combinatorial class and let $\leq$ be a partial order on $\mathcal{C}$. We say that $(\mathcal{C},\leq)$ is an ordered combinatorial class if for all $x,y\in\mathcal{C}$, $$x<y\Longrightarrow|x|<|y|$$

Let $x\in\mathcal{C}$ and let $\lambda$ be a finite multiset of objects in $\mathcal{C}$. We say that $\lambda$ is a partition of $x$ if

1. $$y\in\lambda\longrightarrow y\leq x$$
2. $$\underset{y\in\lambda}\sum|y|=|x|$$

Is there literature on these or similar notions?