Related papers: Decidability in geometric grid classes of permutat…

A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods,…

We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of…

We give a bijection between meanders and special sorts of Gauss diagrams which aroused from the Thurston generators of braid groups. It allows us to give an algorithm to construct such diagrams and to code meanders by matrices which are…

Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of…

We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating…

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric…

This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a…

We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and…

We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of…

In a previous work we introduced an elementary method to analyze the periodicity of a generating function defined by a single equation y=G(x,y). This was based on deriving a single set-equation Y = Gammma(Y) defining the spectrum of the…

We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the…

Let G be the space of generating functions of a periodic infinite order linear recurrence. In this paper we provide an explicit procedure for computing a basis of G.

The structure of three pattern classes Av(2143, 4321), Av(2143, 4312) and Av(1324, 4312) is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams…

In this paper, we present an algorithm that enumerates a certain class of signed permutations, referred to as grid signed permutation classes. In the case of permutations, the corresponding grid classes are of interest because they are…

We show that the change of basis matrices of a set of $m$ bases of a finite vector space is a connected groupoid of order $m^2$. We define a general method to express the elements of change of basis matrices as algebraic expressions using…

We develop a rewriting theory suitable for diagrammatic algebras and lay down the foundations of a systematic study of their higher structures. In this paper, we focus on the question of finding bases. As an application, we give the first…

A notion of branch-width, which generalizes the one known for graphs, can be defined for matroids. We first give a proof of the polynomial time model-checking of monadic second-order formulas on representable matroids of bounded…

Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of…