限制栈的栈排序
Stack sorting with restricted stacks
摘要：
（经典）问题的特点和枚举排列，可以排序使用两个堆栈串联连接仍然是开放的。在本文中，我们解决一个相关的问题，在其中，我们施加限制的程序和堆栈。更准确地说，我们考虑一个贪心算法，我们执行最右边的合法操作（这里的“最右边”是指堆栈排序问题的通常表示）。此外，对于某个置换$σ$，第一堆栈需要是$σ$避免的，这意味着，在每一步，当从顶部到底部读取时，堆栈中维护的元素避免模式$σ$。由于可以被这样一个设备（我们称之为$σ$ -machine）排序的排列集合并不总是一个类，所以理解它何时发生是很有趣的。我们将证明，一组$σ$ -机器，其相关的可排序排列是不是一个类是由加泰罗尼亚数计数。此外，我们将分析两个特定的$σ$ -机器的全部细节（即当$σ=321$和$σ=123$），为他们每个人提供一个完整的表征和枚举的排序排列。
英文摘要：
The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $σ$-avoiding, for some permutation $σ$, meaning that, at each step, the elements maintained in the stack avoid the pattern $σ$ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call $σ$-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of $σ$-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific $σ$-machines in full details (namely when $σ=321$ and $σ=123$), providing for each of them a complete characterization and enumeration of sortable permutations.
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