A Decreasing Stack and an Increasing Stack in Series

TL;DR

This paper analyzes a specialized sorting machine with two stacks in series, where the first stack must be decreasing, and characterizes the permutations it can sort, showing they are counted by Schr"oder numbers.

Contribution

It introduces a new stack sorting model with a decreasing stack in series and provides a basis for the sortable permutations, linking it to Schr"oder numbers.

Findings

Sortable permutations are enumerated by Schr"oder numbers

The basis of the permutation class is characterized

The model extends classical stack sorting analysis

Abstract

We study a sorting machine consisting of two stacks in series where the first stack has the added restriction such that entries in the stack must be in decreasing order from top to bottom. We give the basis of the class of permutations that are sortable by this machine which shows that it is enumerated by the Schr\"oder numbers.