Let $\mathbb N=\{1,2,3,\dots\}$. Let $(P,\le_P)$ be a nonempty finite poset with $n=|P|\ge 1$, and let $\omega:P\to \{1,\dots,n\}$ be a bijection. For a positive integer $m$, let $[m]=\{1,\dots,m\}$. A bounded $(P,\omega)$-partition is a map $\sigma:P\to [m]$ such that, for all $x,y\in P$ with $x\le_P y$, one has $\sigma(x)\le \sigma(y)$, and if additionally $x<_P y$ and $\omega(x)>\omega(y)$, then $\sigma(x)<\sigma(y)$. Let $\mathcal L(P,\omega)$ be the set of words $\pi=(\pi_1,\dots,\pi_n)$ obtained by listing the labels of the elements of $P$ in an order compatible with $\le_P$. For such a word, define $\operatorname{Des}(\pi)=\{i\in\{1,\dots,n-1\}:\pi_i>\pi_{i+1}\}$ and $\operatorname{des}(\pi)=|\operatorname{Des}(\pi)|$. Then the number of bounded $(P,\omega)$-partitions is

\begin{align*} \sum_{\pi\in \mathcal L(P,\omega)} \binom{m+n-1-\operatorname{des}(\pi)}{n}. \end{align*}

The [binomial coefficient](/page/Binomial%20Coefficient) is interpreted as $0$ when its upper parameter is less than $n$.