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. Moreover, the first stack is required to be σ-avoiding, for some permutation σ, meaning that 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 a σ-machine, is not always a class, it would be interesting to understand when it happens. We prove that the set of σ-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we analyze two specific σ-machines in full detail (namely $\sigma =321$ and $\sigma =123$), providing for each of them a complete characterization and enumeration of the sortable permutations.

可以使用串联连接的两个堆栈进行排序的表征和枚举排列的（经典）问题在很大程度上仍未解决。在本文中，我们解决了一个相关的问题，我们在过程和堆栈上都施加了限制。更准确地说，我们考虑一种贪婪算法，在其中执行最正确的合法操作。另外，第一叠层需要是σ -avoiding，对于一些置换σ，这意味着该元件在堆栈避免保持图案σ当读取从顶部到底部。由于可以通过这种设备排序的排列集，我们称为σ机器，它并不总是一类，了解它何时发生会很有趣。我们证明，其关联的可排序排列不是一类的σ-机器集由加泰罗尼亚数计算。此外，我们将详细分析两个特定的σ机器（即$\sigma =321$ 和 $\sigma =123$），为它们中的每一个提供完整的特征描述和可排序排列的枚举。