On permutation classes defined by token passing networks, gridding matrices and pictures : three flavours of involvement

Abstract

The study of pattern classes is the study of the involvement order on finite permutations. This order can be traced back to the work of Knuth. In recent years the area has attracted the attention of many combinatoralists and there have been many structural and enumerative developments. We consider permutations classes defined in three different ways and demonstrate that asking the same fixed questions in each case motivates a different view of involvement. Token passing networks encourage us to consider permutations as sequences of integers; grid classes encourage us to consider them as point sets; picture classes, which are developed for the first time in this thesis, encourage a purely geometrical approach. As we journey through each area we present several new results.

We begin by studying the basic definitions of a permutation. This is followed by a discussion of the questions one would wish to ask of permutation classes. We concentrate on four particular areas: partial well order, finite basis, atomicity and enumeration. Our third chapter asks these questions of token passing networks; we also develop the concept of completeness and show that it is decidable whether or not a particular network is complete. Next we move onto grid classes, our analysis using generic sets yields an algorithm for determining when a grid class is atomic; we also present a new and elegant proof which demonstrates that certain grid classes are partially well ordered.

The final chapter comprises the development and analysis of picture classes. We completely classify and enumerate those permutations which can be drawn from a circle, those which can be drawn from an X and those which can be drawn from some convex polygon. We exhibit the first uncountable set of closed classes to be found in a natural setting; each class is drawn from three parallel lines. We present a permutation version of the famous `happy ending' problem of Erdös and Szekeres. We conclude with a discussion of permutation classes in higher dimensional space.