# Journal of Combinatorics

## Volume 10 (2019)

### Special Issue in Memory of Jeff Remmel, Part 1 of 2

Guest Editor: Nicholas A. Loehr

### Patterns in words of ordered set partitions

Pages: 433 – 490

DOI: https://dx.doi.org/10.4310/JOC.2019.v10.n3.a2

#### Authors

Dun Qiu (Department of Mathematics, University of California at San Diego)

Jeffrey Remmel (Department of Mathematics, University of California at San Diego)

#### Abstract

An ordered set partition of $\lbrace 1, 2, \dotsc, n \rbrace$ 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 $\lvert \mathcal{OP}_{n,k}(\sigma) \rvert$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $\lvert \mathcal{OP}_{n,k}(\sigma) \rvert$ 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 $\lvert \mathcal{WOP}_{n,k}(\sigma) \rvert$ 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 $3$.

#### Keywords

permutations, ordered set partitions, pattern avoidance, bijections, Dyck paths

#### 2010 Mathematics Subject Classification

Primary 05A15, 05A18. Secondary 05A05, 05A10, 05A19.