Adeline Pierrot

2-Stack Sorting is Polynomial

In this article, we give a polynomial algorithm to decide whether a given permutation σ is sortable with two stacks in series. This is indeed a longstanding open problem which was first introduced by Knuth ([1973]). 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 relies on Pierrot and Rossin ([2013]) in which 2-stack pushall sorting is defined and studied.

An algorithm computing combinatorial specifications of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.

2-Stack Sorting is polynomial.

This article deals with deciding whether a permutation is sortable with two stacks in series. Whether this decision problem lies in P or is NP-complete is a longstanding open problem since the introduction of serial compositions of stacks by Knuth in The Art of Computer Programming in 1973. We hereby prove that this decision problem lies in P by giving a polynomial algorithm to solve it. This algorithm uses the concept of pushall sorting, which was previously defined and studied by the authors.