Rationality for subclasses of 321-avoiding permutations

We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is in bijective correspondence with a regular language. The proof makes significant use of formal languages and of a host of encodings, including a new mapping called the panel encoding that maps languages over the infinite alphabet of positive integers avoiding certain subwords to languages over finite alphabets

Prolific structures in combinatorial classes

Under what circumstances might every extension of a combinatorial structure contain more copies of another one than the original did? This property, which we call prolificity, holds universally in some cases (e.g., finite linear orders) and only trivially in others (e.g., permutations). Integer compositions, or equivalently layered permutations, provide a middle ground. In that setting, there are prolific compositions for a given pattern if and only if that pattern begins and ends with 1. For each pattern, there are methods that identify conditions that allow classification of the texts that are prolific for the pattern. This notion is also extendable to other combinatorial classes. In the context of permutations that are sums of cycles we can also establish minimal elements for the set of prolific permutations based on the bijective correspondence between these permutations and compositions, with a slightly different containment order. We also take a brief step into the more general world of permutations that avoid the pattern 321 and attempt to establish some preliminary results

Rationality for subclasses of 321-avoiding permutations

We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well-quasi-ordered has a rational generating function. To do so we show that any such class is in bijective correspondence with a regular language. The proof makes significant use of formal languages and of a host of encodings, including a new mapping called the panel encoding that maps languages over the infinite alphabet of positive integers avoiding certain subwords to languages over finite alphabets

Minimal Classes of Unbounded Clique-Width

In the study of graphs, clique-width is a parameter that has received much attention due its significance in the tractability of algorithms on certain classes of graph. Of particular interest are hereditary graph classes, those classes closed under taking induced subgraphs. A number of minimal hereditary graph classes of unbounded clique-width (abbreviated to minimal classes) have recently been identified; that is, classes containing graphs with arbitrarily large clique-width but where every proper hereditary subclass has bounded clique-width. There are also hereditary classes of unbounded clique-width that do not contain a minimal subclass, but instead contain graph structures known as t-basic obstructions to bounded clique-width for arbitrarily large t. These graphs form a sequence known as an antichain of unbounded clique-width. We identify many new minimal classes and place all known minimal classes inside two `frameworks'. We also identify new t-basic obstructions to bounded clique-width. In Chapters 2 and 3 we create our first framework for dense minimal classes, consisting of graph classes constructed by taking the finite induced subgraphs of an infinite graph Pδ whose vertices form a two-dimensional array and whose edges are defined by three objects, denoted as a triple δ =(α, β, γ). We introduce new methods to the study of clique-width, and identify uncountably many new minimal classes in the framework. In sparse classes clique-width is unbounded if and only if the (widely studied) parameter, tree-width, is unbounded. In Chapter 4 we identify a new t-basic obstruction, a t-sail. We construct `path-star' graph classes defined by a nested word, with a recursive structure, in which a graph has large tree-width if and only if it contains a large t-sail. We show that these classes are infinitely defined and do not contain a minimal subclass. In Chapter 5 we create an alternative framework for minimal classes to the one developed in Chapters 2 and 3, containing `path-clique' graph classes consisting of the finite induced subgraphs of an infinite graph created from the symmetric difference of edges between an infinite path and a partition of the path vertices forming infinite cliques or independent sets that are complete or anti-complete to each other. We identify another uncountable family of minimal classes different to those from the first framework. In Chapter 6 we identify a new t-basic obstruction -- a t-clipper. We show that a graph in the class of permutation-partition graphs has large clique-width if and only if it contains a large t-clipper. We also identify other likely t-basic obstructions