Permutation classes

This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics

Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons

The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < \alpha_*(p) \leq \beta^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $\alpha_*(p) \approx 0.812$ and $\beta^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $\alpha_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{\alpha_*(p) - o(1)}$ and $n^{\beta^*(p) + o(1)}$ (resp. $n^{\alpha_*(1-p) - o(1)}$ and $n^{\beta^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.Comment: New version before journal submissio

Scaling limits of permutations avoiding long decreasing sequences

We determine the scaling limit for permutations conditioned to have longest decreasing subsequence of length at most $d$. These permutations are also said to avoid the pattern $(d+1)d \cdots 2 1$ and they can be written as a union of $d$ increasing subsequences. We show that these increasing subsequences can be chosen so that, after proper scaling, and centering, they converge in distribution. As the size of the permutations tends to infinity, the distribution of functions generated by the permutations converges to the eigenvalue process of a traceless $d\times d$ Hermitian Brownian bridge.Comment: 58 pages, 10 figures, minor edits and additional figures adde

The enumeration of subclasses of the 321-avoiding permutations

This thesis is dedicated to the enumeration of subclasses of 321-avoiding permutations, using a combination of theoretical and experimental investigations. The thesis is organised as follows: Chapter 1 provides the necessary definitions and preliminaries, discusses the current state of research on the subject of enumerating Av(321) and its subclasses, then gives an introduction on the basic problem of containment check for 321-avoiding permutations, the process of which is used throughout our work. The main results of this study are explained in Chapter 2 and 3. Chapter 2 focuses on the implementation aspects of enumerating 321-avoiding classes, where the main goal is to develop efficient algorithms to generate all permutations up to a certain length contained in classes of the form Av(321, π). The permutation counts are then used to guess the generating function by fitting a rational function to the computed data. In Chapter 3, we deal with the more theoretical problem of enumerating 321-avoiding polynomial classes given a structural description. In particular, we propose a method which computes the grid class of such a class given its basis. We then use this information to enumerate the class using an improved version of a known algorithm

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polynomials, we describe a bijection between the class of k-parallelogram polyominoes and the class of planted planar trees having height less than k+3. In the second part of the thesis we examine the notion of pattern avoidance, which has been extensively studied for permutations. We introduce the concept of pattern avoidance in the context of matrices, more precisely permutation matrices and polyomino matrices. We present definitions analogous to those given for permutations and in particular we define polyomino classes, i.e. sets downward closed with respect to the containment relation. So, the study of the old and new properties of the redefined sets of objects has not only become interesting, but it has also suggested the study of the associated poset. In both approaches our results can be used to treat open problems related to polyominoes as well as other combinatorial objects.Comment: PhD thesi

Aspects of Metric Spaces in Computation

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Detectando agrupamientos y contornos: un estudio doble sobre representación de formas

Las formas juegan un rol clave en nuestro sistema cognitivo: en la percepción de las formas yace el principio de la formación de conceptos. Siguiendo esta línea de pensamiento, la escuela de la Gestalt ha estudiado extensivamente la percep- ción de formas como el proceso de asir características estructurales encontradas o impuestas sobre el material de estímulo.En resumen, tenemos dos modelos de formas: pueden existir físicamente o ser un producto de nuestros procesos cogni- tivos. El primer grupo está compuesto por formas que pueden ser definidas extra- yendo los contornos de objetos sólidos. En este trabajo nos restringiremos al caso bidimensional. Decimos entonces que las formas del primer tipo son formas planares. Atacamos el problema de detectar y reconocer formas planares. Cier- tas restricciones teóricas y prácticas nos llevan a definir una forma planar como cualquier pedazo de línea de nivel de una imagen. Comenzamos por establecer que los métodos a contrario existentes para de- tectar líneas de nivel son usualmente muy restrictivos: una curva debe ser enter- amente saliente para ser detectada. Esto se encuentra en clara contradicción con la observación de que pedazos de líneas de nivel coinciden con los contornos de los objetos. Por lo tanto proponemos una modificación en la que el algoritmo de detección es relajado, permitiendo la detección de curvas parcialmente salientes. En un segundo acercamiento, estudiamos la interacción entre dos maneras diferentes de determinar la prominencia de una línea de nivel. Proponemos un esquema para competición de características donde el contraste y la regularidad compiten entre ellos, resultando en que solamente las líneas de nivel contrastadas y regulares son consderedas salientes. Una tercera contribución es un algoritmo de limpieza que analiza líneas de nivel salientes, descartando los pedazos no salientes y conservando los salientes. Está basado en un algoritmo para detección de multisegmentos que fue extendido para trabajar con entradas periódicas. Finalmente, proponemos un descriptor de formas para codificar las formas detectadas, basado en el Shape Context global. Cada línea de nivel es codificada usando shape contexts, generando así un nuevo descriptor semi-local. A contin- uación adaptamos un algoritmShape plays a key role in our cognitive system: in the perception of shape lies the beginning of concept formation. Following this lines of thought, the Gestalt school has extensively studied shape perception as the grasping of structural fea- tures found in or imposed upon the stimulus material. In summary, we have two models for shapes: they can exist physically or be a product of our cognitive pro- cesses. The first group is formed by shapes that can be defined by extracting contours from solid objects. In this work we will restrict ourselves to the two dimensional case. Therefore we say that these shapes of the first type are planar shapes. We ad- dress the problem of detecting and recognizing planar shapes. A few theoretical and practical restrictions lead us to define a planar shape as any piece of mean- ingful level line of an image. We begin by stating that previous a contrario methods to detect level lines are often too restrictive: a curve must be entirely salient to be detected. This is clearly in contradiction with the observation that pieces to level lines coincide with object boundaries. Therefore we propose a modification in which the detection criterion is relaxed by permitting the detection of partially salient level lines. As a second approach, we study the interaction between two different ways of determining level line saliency: contrast and regularity. We propose a scheme for feature competition where contrast and regularity contend with each other, resulting in that only contrasted and regular level lines are considered salient. A third contribution is a clean-up algorithm that analyses salient level lines, discarding the non-salient pieces and returning the salient ones. It is based on an algorithm for multisegment detection, which was extended to work with periodic inputs. Finally, we propose a shape descriptor to encode the detected shapes, based on the global Shape Context. Each level line is encoded using shape contexts, thus generating a new semi-local descriptor. We then adapt an existing a contrario shape matching algorithm to our particular case. The second group is composed by shapes that do not correspond to a solid object but are formed by integrating several solid objects. The simplest shapes in this group are arrangements of points in two dimensions. Clustering techniques might be helpful in these situations. In a seminal work from 1971, Zahn faced the problem of finding perceptual clusters according to the proximity gestalt and proposed three basic principles for clustering algorithms: (1) only inter-point distances matter, (2) stable results across executions and (3) independence from the exploration strategy. A last implicit requirement is crucial: clusters may have arbitrary shapes and detection algorithms must be capable of dealing with this. In this part we will focus on designing clustering methods that completely fulfils the aforementioned requirements and that impose minimal assumptions on the data to be clustered. We begin by assessing the problem of validating clusters in a hierarchical struc- ture. Based on nonparametric density estimation methods, we propose to com- pute the saliency of a given cluster. Then, it is possible to select the most salient clusters in the hierarchy. In practice, the method shows a preference toward com- pact clusters and we propose a simple heuristic to correct this issue. In general, graph-based hierarchical methods require to first compute the com- plete graph of interpoint distances. For this reason, hierarchical methods are often considered slow. The most usually used, and the fastest hierarchical clustering al- gorithm is based on the Minimum Spanning Tree (MST). We therefore propose an algorithm to compute the MST while avoiding the intermediate step of computing the complete set of interpoint distances. Moreover, the algorithm can be fully par- allelized with ease. The algorithm exhibits good performance for low-dimensional datasets and allows for an approximate but robust solution for higher dimensions. Finally we propose a method to select clustered subtrees from the MST, by computing simple edge statistics. The method allows naturally to retrieve clus- ters with arbitrary shapes. It also works well in noisy situations, where noise is regarded as unclustered data, allowing to separate it from clustered data. We also show that the iterative application of the algorithm allows to solve a phenomenon called masking, where highly populated clusters avoid the detection less popu- lated ones.Fil:Tepper, Mariano. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina