论文标题
$ K $块设置分区模式和统计数据
$K$ block set partition patterns and statistics
论文作者
论文摘要
$ [n] = \ {1，\ cdots，n \} $的设置分区$σ$，如果对子集$ s \ subseteq [n] $的标准化限制为$σ$，则包含另一个集合分区$ω$。否则，$σ$避免了$ω$。萨根（Sagan）和戈特（Goyt）确定了$ [3] $分区中所有图案的回避类别的基数。此外，设定的分区与受限生长功能（RGF）之间存在培训。 Wachs和White定义了这些RGF的四个基本统计数据。 Sagan，Dahlberg，Dorward，Gerhard，Grubb，Purcell和Reppuhn考虑了这些统计数据在各种回避类别上的分布，它们获得了先前引用的基数结果的四个变化类似物。他们对这些分布进行了首次彻底研究。他们的许多结果的类似物是针对固定分区的，恰好是$ k $块，用于指定的正整数$ k $。这些类似物在这项工作中进行了讨论。
A set partition $σ$ of $[n]=\{1,\cdots ,n\}$ contains another set partition $ω$ if a standardized restriction of $σ$ to a subset $S\subseteq[n]$ is equivalent to $ω$. Otherwise, $σ$ avoids $ω$. 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.
