A Bijection on Classes Enumerated by the Schröder Numbers

We consider a sorting machine consisting of two stacks in series where thefirst stack has the added restriction that entries in the stack must be indecreasing order from top to bottom. The class of permutations sortable by thismachine are known to be enumerated by the Schröder numbers. In this paper, wegive a bijection between these sortable permutations of length $n$ andSchröder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed ofEast steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travelweakly below the line $y=x$