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

On The Möbius Function Of Permutations Under The Pattern Containment Order

We study several aspects of the Möbius function, μ[σ, π], on the poset of permutations under the pattern containment order. First, we consider cases where the lower bound of the poset is indecomposable. We show that μ[σ, π] can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities. We then consider conditions on an interval which guarantee that the value of the Möbius function is zero. In particular, we show that if a permutation π contains two intervals of length 2, which are not order-isomorphic to one another, then μ[1, π] = 0. This allows us to prove that the proportion of permutations of length n with principal Möbius function equal to zero is asymptotically bounded below by (1−1/e) 2 ≥ 0.3995. This is the first result determining the value of μ[1, π] for an asymptotically positive proportion of permutations π. Following this, we use “2413-balloon” permutations to show that the growth of the principal Möbius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial. We then generalise 2413-balloon permutations, and find a recursion for the value of the principal Möbius function of these generalisations. Finally, we look back at the results found, and discuss ways to relate the results from each chapter. We then consider further research avenues

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group