  In this article, we give a polynomial algorithm to decide whether a given
permutation $\sigma$ is sortable with two stacks in series. This is indeed a
longstanding open problem which was first introduced by Knuth. He introduced
the stack sorting problem as well as permutation patterns which arises
naturally when characterizing permutations that can be sorted with one stack.
When several stacks in series are considered, few results are known. There are
two main different problems. The first one is the complexity of deciding if a
permutation is sortable or not, the second one being the characterization and
the enumeration of those sortable permutations. We hereby prove that the first
problem lies in P by giving a polynomial algorithm to solve it. This article
strongly relies on a previous article in which 2-stack pushall sorting is
defined and studied.
