Compression of Meanders

This paper refers to the algorithmic transformation of a meander to its uniquely defined compression. We obtain this directly from meandric permutations, thus creating representations of large classes of meanders of different orders. We prove basic properties, give arithmetic results, and produce generating procedures. 1. Introduction A closed meander of order is a closed self-avoiding curve crossing an infinite horizontal line times [1]. In this paper, we obtain the compression as the determination of a unique simple meander, directly from its permutation. The meanders as planar permutations were introduced by Rosenstiehl [2] and they have been studied with nested sets [3, 4]. More specifically, in Section 2, we define the flow of a meander consisted by its traces and corresponding blocks. In Section 3, we create a specific form of meanders: the simple ones, we study the properties of their numbers of cuttings and cutting degree and we use them in order to introduce the compression. In Section 4, we determine the flow of the meandric permutations and we achieve also numerical results for the classification of the meanders of the compressions according to their order. Finally, in Section 5, we establish the compression of meanders directly from their meandric permutations divided in suitable blocks. Thus, we change their interpretation and produce a simplified procedure for generating the compressions. The following definitions and notation are necessary for the rest of the paper [3]. A set of disjoint pairs of such that and for any we never have is called nested set of pairs on . Each pair of a nested set consists of an odd and an even number. We denote the set of all nested sets of pairs on by . Two nested sets define a permutation on , such that and , for every . The sets are -matching if and only if has cycles. In the case where , are simply called matching. This definition is equivalent to the one given in [3]. We call short pair of any pair of consecutive numbers that belongs to , and outer pair of any pair such that there is no pair with . Each nested set of pairs contains at least one outer and one short pair. 2. Meanders A meander of order is equivalently defined [3] as a cyclic permutation on , for which the following properties hold true: , and the sets are both nested and matching. We take all numbers . It is clear that is odd if and only if is odd. In the corresponding geometrical representation, the nested arcs correspond to nested pairs. A pair of nested sets , should be matching, in order to generate a meander. For example, the meander