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A Bijection on Classes Enumerated by the Schröder Numbers
Discrete mathematics & theoretical computer science, Permutation Patterns 2015, Tome 18 (2015-2016) no. 2.

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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 Schröder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder 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 $y=x$. 
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     author = {Schroeder, Michael W. and Smith, Rebecca},
     title = {A {Bijection} on {Classes} {Enumerated} by the {Schr\"oder} {Numbers}},
     journal = {Discrete mathematics & theoretical computer science},
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Schroeder, Michael W.; Smith, Rebecca. A Bijection on Classes Enumerated by the Schröder Numbers. Discrete mathematics & theoretical computer science, Permutation Patterns 2015, Tome 18 (2015-2016) no. 2. doi : 10.46298/dmtcs.1326. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1326/

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