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 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 $y=x$.
Highlights
A stack is a sorting device that works by a sequence of push and pop operations
The class of stack-sortable permutations is enumerated by the Catalan numbers
For every Schroder path S from (0, 0) to (n − 1, n − 1), there is an ADI word W of a DI-sortable permutation of length n such that Algorithm 3.2 applied to W gives S
Summary
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 is known to be enumerated by the Schroder numbers. We give a bijection between these sortable permutations of length n and Schroder paths of order n − 1: 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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