• $B_1 \cup B_2 \cup \dots \cup B_k = X$.
• $B_i \cap B_j = \emptyset$ if $i \neq j$.
(a) List all ordered partitions of the set $[3] = \{1, 2, 3\}$.
(b) Let $c_k(n)$ be the number of ordered partitions of $[n]$ with $k$ blocks. Prove that $c_k(n) = k!S_{n,k}$, where $S_{n,k}$ is the Stirling number of the second kind.
(c) Fix $k$. Find the exponential generating function of $(c_k(n))_{n \geq 0}$.
(d) Let $c(n)$ be the number of all ordered partitions on $[n]$. Let $c(0) = 1$, Find the exponential generating function of $(c(n))_{n \geq 0}$.
