Patterns in words of ordered set partitions

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(\sigma)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $\sigma$ and obtained the formula for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $3$. In this paper, we define a new pattern avoidance for the set $\mathcal{OP}_{n,k}$, called $\mathcal{WOP}_{n,k}(\sigma)$, which includes the questions proposed by Godbole, Goyt, Herdan and Pudwell. We obtain formulas for $|\mathcal{WOP}_{n,k}(\sigma)|$ combinatorially for any $\sigma$ of length $3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $\mathcal{WOP}_{n,k}(\sigma)$ for $\sigma$ of length $3$.