Longest Common Pattern between two Permutations

In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations

Longest Common Separable Pattern between Permutations

In this article, we study the problem of finding the longest common separable pattern between several permutations. We give a polynomial-time algorithm when the number of input permutations is fixed and show that the problem is NP-hard for an arbitrary number of input permutations even if these permutations are separable. On the other hand, we show that the NP-hard problem of finding the longest common pattern between two permutations cannot be approximated better than within a ratio of $sqrt{Opt}$ (where $Opt$ is the size of an optimal solution) when taking common patterns belonging to pattern-avoiding classes of permutations.Comment: 15 page

The Longest Common Pattern Problem for two Permutations

International audienceIn this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations

Alternating, pattern-avoiding permutations

We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration of the latter set to that of the former.Comment: 7 page

Simple permutations poset

This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are two simple permutations such that $\pi < \sigma$ then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, ..., \sigma^{(k)}=\pi$ such that $|\sigma^{(i)}| - |\sigma^{(i+1)}| = 1$ - or 2 when permutations are exceptional- and $\sigma^{(i+1)} < \sigma^{(i)}$. This characterization induces an algorithm polynomial in the size of the output to compute the simple permutations in a wreath-closed permutation class.Comment: 15 page