$K$ block set partition patterns and statistics

TL;DR

This paper extends the study of set partition pattern avoidance by analyzing distributions of fundamental statistics on restricted growth functions, focusing on partitions with a fixed number of blocks and providing new combinatorial insights.

Contribution

It generalizes previous avoidance and distribution results to set partitions with exactly k blocks, enriching the combinatorial understanding of pattern avoidance.

Findings

Distribution formulas for set partitions with k blocks

Bijections between set partitions and restricted growth functions

Extended enumeration results for avoidance classes

Abstract

A set partition $\sigma$ of $[n]=\{1,\cdots ,n\}$ contains another set partition $\omega$ if a standardized restriction of $\sigma$ to a subset $S\subseteq[n]$ is equivalent to $\omega$. Otherwise, $\sigma$ avoids $\omega$. Sagan and Goyt have determined the cardinality of the avoidance classes for all sets of patterns on partitions of $[3]$. Additionally, there is a bijection between the set partitions and restricted growth functions (RGFs). Wachs and White defined four fundamental statistics on those RGFs. Sagan, Dahlberg, Dorward, Gerhard, Grubb, Purcell, and Reppuhn consider the distributions of these statistics over various avoidance classes and they obtained four variate analogues of the previously cited cardinality results. They did the first thorough study of these distributions. The analogues of their many results follows for set partitions with exactly $k$ blocks for a specified positive integer $k$. These analogues are discussed in this work.