论文标题
高度分类的排列和钟数
Highly Sorted Permutations and Bell Numbers
论文作者
论文摘要
令$ s $表示韦斯特的堆栈分类地图。对于所有正整数$ m $和所有整数$ n \ geq 2m-2 $，我们给出了集合$ s^{n-m}（n-m}（s_n）$的简单表征;结果，我们发现$ | s^{n-m}（s_n）| $是$ m^\ text {th} $ bell number $ b_m $。我们还证明，通过显示$ | s^{m-3}（s_ {2m-3}）的限制$ n \ geq 2m-2 $是紧密的，所有$ m \ geq 3 $。
Let $s$ denote West's stack-sorting map. For all positive integers $m$ and all integers $n\geq 2m-2$, we give a simple characterization of the set $s^{n-m}(S_n)$; as a consequence, we find that $|s^{n-m}(S_n)|$ is the $m^\text{th}$ Bell number $B_m$. We also prove that the restriction $n\geq 2m-2$ is tight by showing that $|s^{m-3}(S_{2m-3})|=B_m+m-2$ for all $m\geq 3$.
