Set partition patterns and statistics

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• 作者：Samantha Dahlberg^a ; ^{dahlbe14@msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Robert Dorward^b ; ^{rdorward@oberlin.edu" class="auth_mail" title="E-mail the corresponding author} ; Jonathan Gerhard^c ; ^{gerha2jm@dukes.jmu.edu" class="auth_mail" title="E-mail the corresponding author} ; Thomas Grubb^a ; ^{grubbtho@msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Carlin Purcell^d ; ^{carlinpurcell@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Lindsey Reppuhn^e ; ^{Lindsey.Reppuhn11@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Bruce E. Sagan^a ; ^{sagan@math.msu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Avoidance ; Fibonacci number ; Generating function ; Integer partition ; lblb ; lsls ; Pattern ; rbrb ; rsrs ; Set partition ; Statistic
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：1-16
• 全文大小：434 K

文摘

A set partition 33b9924fc2805691cdde" title="Click to view the MathML source">σ$\sigma$ of [n]={1,…,n}$\left[n\right]=\{1,\dots ,n\}$ contains another set partition π$\pi$ if restricting 33b9924fc2805691cdde" title="Click to view the MathML source">σ$\sigma$ to some S⊆[n]$S\subseteq \left[n\right]$ and then standardizing the result gives π$\pi$. Otherwise we say 33b9924fc2805691cdde" title="Click to view the MathML source">σ$\sigma$ avoids π$\pi$. For all sets of patterns consisting of partitions of [3], the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previously cited cardinality results. This is the first in-depth study of such distributions. We end with a list of open problems.