Related papers: On the sub-permutations of pattern avoiding permut…
Graphs are extremely versatile and ubiquitous mathematical structures with potential to model a wide range of domains. For this reason, graph problems have been of interest since the early days of computer science. Some of these problems…
In this article, we describe an algorithm to determine whether a permutation class C given by a finite basis B of excluded patterns contains a finite number of simple permutations. This is a continuation of the work initiated in [Brignall,…
Given two permutations $\sigma$ and $\pi$, the \textsc{Permutation Pattern} problem asks if $\sigma$ is a subpattern of $\pi$. We show that the problem can be solved in time $2^{O(\ell^2\log \ell)}\cdot n$, where $\ell=|\sigma|$ and…
We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$. First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(\pi)|$ for any permutation $\pi$, answering a question of…
Given a set $Y$ of decreasing plane trees and a permutation $\pi$, how many trees in $Y$ have $\pi$ as their postorder? Using combinatorial and geometric constructions, we provide a method for answering this question for certain sets $Y$…
We extend classical results on simple varieties of trees (asymptotic enumeration, average behavior of tree parameters) to trees counted by their number of leaves. Motivated by genome comparison of related species, we then apply these…
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular…
Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any…
We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{\dh}mundsson. Our…
Applying a method to reconstruct a phylogenetic tree from random data provides a way to detect whether that method has an inherent bias towards certain tree `shapes'. For maximum parsimony, applied to a sequence of random 2-state data, each…
Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying…
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between…
A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a…
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this…
Every word has a shape determined by its image under the Robinson-Schensted-Knuth correspondence. We show that when a word w contains a separable (i.e., 3142- and 2413-avoiding) permutation \sigma\ as a pattern, the shape of w contains the…
We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.
This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns in Permutations and words}. I first survey recent developments on the enumeration…
Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a $d$-dimensional generalization of these measures for all…
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following \cite{BCS}, let $e_k\pi$…
We establish a bijection from the set of all permutations (of a given length) that avoid a pattern $q$ and a context-sensitive language.