We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern.
The algorithm provides a new proof of the
description of West-2-stack-sortable permutations, that is
permutations that are completely sorted when passed twice through a
stack, in terms of patterns. We also
solve the long-standing problem of describing
West-3-stack-sortable permutations. This requires a new type of
generalized permutation pattern we call a decorated pattern.
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