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Ordered Partitions Avoiding a Permutation of Length 3

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 $\mathcal{OP}_{n,k}(σ)$ be set of ordered partitions in $\mathcal{OP}_{n,k}$ that avoid a pattern $σ$. 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 $|\mathcal{OP}_{n,k}(σ)|=|\mathcal{OP}_{n,k}(123)|$ for any permutation $σ$ 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.