A Bijection on Classes Enumerated by the Schr\"oder Numbers

TL;DR

This paper establishes a bijection between permutations sortable by a specific two-stack machine and Schr"oder paths, providing a combinatorial link that explains their shared enumeration by Schr"oder numbers.

Contribution

It introduces a new bijection connecting sortable permutations with Schr"oder paths, clarifying their combinatorial structure and enumeration.

Findings

Permutations sortable by the machine are counted by Schr"oder numbers.

A bijection between these permutations and Schr"oder paths is constructed.

The bijection provides combinatorial insight into the enumeration.

Abstract

We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schr\"oder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schr\"oder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.