  An ordered partition of $[n]=\{1, 2, \ldots, n\}$ is a partition whose blocks
are endowed with a linear order. Let $\mathcal{OP}_{n,k}$ be set of ordered
partitions of $[n]$ with $k$ blocks and $\mathcal{OP}_{n,k}(\sigma)$ be set of
ordered partitions in $\mathcal{OP}_{n,k}$ that avoid a pattern $\sigma$.
Recently, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of
ordered partitions of $[n]$ with 3 blocks and the number of ordered partitions
of $[n]$ with $n-1$ blocks avoiding a permutation pattern of length 3. They
showed that $|\mathcal{OP}_{n,k}(\sigma)|=|\mathcal{OP}_{n,k}(123)|$ for any
permutation $\sigma$ of length 3, and raised the question concerning the
enumeration of $\mathcal{OP}_{n,k}(123)$. They also conjectured that the number
of ordered partitions of $[2n]$ with blocks of size 2 avoiding a permutation
pattern of length 3 satisfied a second order linear recurrence relation. In
answer to the question of Godbole, et al., we obtain the generating function
for $|\mathcal{OP}_{n,k}(123)|$ and we prove the conjecture on the recurrence
relation.
