Patterns in words of ordered set partitions

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