Assume $ K $ labeled blocks $ s_1, s_2, \dots, s_K $ ($ s_1 < s_2 < \dots < s_K $) that arrive sequentially and need to be accomodated as they arrive in $ N $ containers (partitions with empty containers also matter as shown in the picture. The capacity of the containers is unlimited.). I need to find an expression that gives all the possible combinations of partitions where the order of the elements matters.
Some examples:
$$
\begin{align}
K = 2, N = 1 \rightarrow & [1 ~ 2] \\
K = 2, N = 2 \rightarrow & [-],[1 ~ 2] \\
& [1 ~ 2],[-] \\
& [1],[2] \\
& [2],[1] \\
K = 2, N = 3 \rightarrow & [-],[-],[1 ~ 2] \\
& [-],[1 ~ 2],[-] \\
& [1 ~ 2],[-],[-] \\
& [-],[2],[1] \\
& [2],[-],[1] \\
& [-],[1],[2] \\
& [2],[1],[-] \\
& [1],[2],[-] \\
& [2],[-],[1]
\end{align}
$$