Patterns in words of ordered set partitions

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.

Full Text (PDF format)

Received 19 April 2018

Published 23 May 2019