Compression of Meanders

A closed meander of order nn is a closed self-avoiding curve crossing an in�nite hori�ontal line 2nn times [1]. In this paper, we obtain the compression as the determination of a unique simplemeander, directly from its permutation.emeanders as planar permutations were introduced by Rosenstiehl [2] and they have been studied with nested sets [3, 4]. More speci�cally, in Section 2, we de�ne the �ow of a meander consisted by its traces and corresponding blocks. In Section 3, we create a speci�c 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 �ow of the meandric permutations andwe achieve also numerical results for the classi�cation 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. us, we change their interpretation and produce a simpli�ed procedure for generating the compressions. e following de�nitions and notation are necessary for the rest of the paper [3]. A set SS of disjoint pairs of [2nnn such that ⋃{aaaaaaaSS{aaa aaa a [2nnn and for any {aaa aaaa {aaa aaa a SS we never have aa a aa a aa a aa is called nested set of pairs on [2nnn. 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 [2nnn byNN2nn. Two nested sets SS1a SS2 a NN2nn de�ne a permutation σσ on [2nnn, such that σσσ2σσ σ 1) a jj iff {2σσ σ 1a jja a SS1 andσσσ2σσ) a jj iff {2σσa jja a SS2, for every σσ a [nnn. e sets SS1a SS2 are kk-matching if and only if σσ has kk cycles. In the case where kk a 1, SS1a SS2 are simply called matching. is de�nition is equivalent to the one given in [3]. We call short pair of SS any pair of consecutive numbers that belongs to SS, and outer pair of SS any pair {aaa aaa a SS such that there is no pair {aaa aaa a SSwith aa a aa a aa a aa. Each nested set of pairs contains at least one outer and one short pair.


Introduction
A closed meander of order  is a closed self-avoiding curve crossing an in�nite hori�ontal line 2 times [1].In this paper, we obtain the compression as the determination of a unique simple meander, directly from its permutation.e meanders as planar permutations were introduced by Rosenstiehl [2] and they have been studied with nested sets [3,4].
More speci�cally, in Section 2, we de�ne the �ow of a meander consisted by its traces and corresponding blocks.In Section 3, we create a speci�c 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 �ow of the meandric permutations and we achieve also numerical results for the classi�cation 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.us, we change their interpretation and produce a simpli�ed procedure for generating the compressions.
e following de�nitions and notation are necessary for the rest of the paper [3].
A set  of disjoint pairs of [2 such that ⋃ { {   [2 and for any {  {    we never have        is called nested set of pairs on [2.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 [2 by  2 .Two nested sets  1   2   2 de�ne a permutation  on [2, such that 2  1)   iff {2  1    1 and 2)   iff {2    2 , for every   [.e sets  1   2 are -matching if and only if  has  cycles.In the case where   1,  1   2 are simply called matching.is de�nition 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.

Meanders
A meander of order  is equivalently de�ned [3] as a cyclic permutation on [2, for which the following properties hold true: 1)  1, and the sets are both nested and matching.We take all numbers mod 2.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 e set of all the meanders of order  is denoted by ℳ  .Let   ℳ  be a meander crossing a horizontal line.Following [5], for any      we consider the vertical line, which shall be called the -line, passing through the middle point of the segment (,    of the horizontal line.e numbers of those arcs of the meandric curve which are intersected by the -line and lie above and beneath the horizontal line of , are called the numbers of cuttings ( and  ′ (, respectively [6].
e sum (  (   ′ ( of the number of those arcs of the meandric curve which are intersected by the -line is called the cutting degree of the meander at .We notice that ( and  ′ ( are of the same parity; hence, ( is always even [5].
e meandric curve always has points of intersection with the -line, which we call traces.Obviously, the number of the traces is equal to (.Starting below the horizontal line, we label the traces with the numbers , , , … , (, knowing that ( (resp.,  ′ () of them are lying above (resp., beneath) the horizontal line.From now on, we will consider that the traces are identical to their corresponding labels.For the meander of Figure 1 and for   , we have (  ,  ′ (  , and (  0.
Beginning from trace 1 and moving clockwise upon the meandric curve, following its "natural �ow, " we obtain a shuffle of the permutation of the traces and the meandric permutation, see Figure 2 where the circled elements are the traces.
In the general case, we have the shuffle with   (  ,   (, … ,   (( being the traces of the meander at  and    ,    , … ,   ( the parts of consecutive elements of the meandric permutation , lying between consecutive traces, called blocks of the meander; that is, the block    ,   (, is the set of the consecutive elements of the permutation , which are lying between the traces   ( and   (  .ese two traces are called the "entrance" trace and the "exit" trace of the block, respectively, while the shuffle   is called �o� of the meander from the trace ( of the -line, or for simplicity -�o�.
For every   (,   (     (   corresponds to the part of the meandric curve starting from the trace   ( and ending at the trace   (  , which we denote by    .If  is odd (resp., even), then this curve lies on the le (resp., right) of the -line.In Figure 1, we denote the curve    by (.
A meander is simple at  if and only if every triple of consecutive terms of its permutation contains elements from both of the sets 1, 2, … , } and   1,   2, … , 2}.
For example, the meander of Figure 3 is simple at   .
We note that a meander can be simple at more than one point.For example, the meander   1   2   is simple at   2 and   .
Let    2 be a meander that is not simple at .Further on, we will study every meander according to its -line, so for simplicity we omit the index  from the notation.Let    1 ,  2 , … ,  2 } be its already de�ned set of blocks, where 2   and the blocks   , for    1  1, , … , 2  1} (resp.,    2  2, , … , 2}), are the ones placed at the le (resp., right) of the -line.Each block     lies between the trace  and the trace   1 of the �ow .
We denote by   ,   2, the closed interval of 2 with ends the traces ,   1.Given a pair of blocks   ,    , ,    1 or ,    2 , with   ⊂   , then the block   is called internal of the block   , and the block   is called external to the block   .We can easily deduce that the number of the internal blocks of a block     is equal to 1/2|    1|  1.
If we replace the blocks   ,   2, of the meander  by blocks having one element (resp., two elements) whenever    1 (resp.,    2), then we obtain a meander  of order   1/2  2   (since its blocks have one or two elements, corresponding to crossing points with the horizontal line) and simple at     1   (counting the points of intersection with the horizontal line to the le of the -line).
e result of the above replacement is the set of blocks    1 ,  has as same invariants with the meander  the traces and the �ow of curves.For example, the compression of the meander  of Figure 1 is the pair ( , where  is the meander of Figure 3 and             .According to the absolute value |(  (  |, we place the elements of ( in decreasing order, while of ( in increasing order.If ( (  , then we correspond the numbers  ′ (   …      to the ordered pairs of (, and the numbers   …   ′ ( to the ordered pairs of (.If (   (resp., (  ), then we correspond to the ordered pairs of ( (resp., () the numbers   …  .It is obvious that the previous numbers coincide with their corresponding traces.

The Flow
In order to start the -�ow  from the �rst trace, we choose the pair {( ( which corresponds to the trace (  .If (  , then this pair is the outer pair of ( with (  odd and less than (.If (  , then this is the pair of ( with the smallest value of |(  (  |. For example, for the meander of Figure �ractically, at �rst we partition the permutation of the meander into classes including the consecutive terms of , which are less or equal to (resp., greater than) .us, we have the partition of  into blocks, putting at the beginning the last remaining elements and marking the traces.
For example, for the meander of Figure 1 we have Figure 4.
e placement of the traces follows the change of parity of the elements of the permutation, if we have an odd (resp., even) element followed by an even (resp., odd) element, then their intermediate trace is lying above (resp., beneath) the hori�ontal line.From the above partition, we obtain the �ow� see Figure 2.
e meanders of the compression of the set ℳ  are partitioned into classes, with elements belonging to the sets In the methods of generating planar permutations [7], we can also include the way to �nd the blocks of meanders, their corresponding numbers   , and consequently the orders of the meanders of their compressions.
ese meanders can be used as generators for the reverse problem of "decompression." We can use them to extend a meander   ℳ  simple at  to all possible meanders   ℳ  , with   .
Table 1 presents the cardinalities of those classes for    3 …  0, without taking into account the corresponding line.
e �eros of the �rst (resp., second) column verify the fact that if the order  of the meander is even (resp., odd), then there do not exist meanders of order  with compression of order 1 (resp., 2).We can easily prove that the values of the �rst column express that there exist |ℳ ( |  meanders of order  with compression of order 1.For meanders of larger order, we should try to calculate the number of different blocks of given orders.

Determining the Compression
We shall �nd the compression of a meander  with the help of its -�ow   (  (  ⋯ (  .We recall that each block    ( consists of one or two elements.Obviously, the elements of   belong to the sets {  …   (resp., {      …  ), when     (resp.,     ).e relative position of these points de�nes a relation of preceding for the blocks of the set (; hence, the block   precedes the block   (  ≺   ), iff min    min   .� Obviously, the above results do not cover the cases of two blocks, the one belonging to the set  2 (, and the other to the set  ′ 2 (, where none of them is internal to the other.In order to obtain a unique solution, we have to make the following choice. (e) If   ∈  2 (,   ∈  ′ 2 (, with min    < min    and   ∈  1 or   ∈  2 , then   ≺   (resp.,   ≺   ), where    =  1   ,  1 is the �rst element of   ,   is the last element of   and  =   .
In the general case, the ordering is deduced from a Hamiltonian path of the directed graphs with vertices the elements of the set  1 (resp.,  2 ) and arcs the pairs (  ∈  2 1 (resp., (  ∈  2  2 ) such that   ≺   .We note that   ≺   iff  −1 ( <  −1 (, given that  −1 ( de�nes the position of the block   at the total order of (.For our example, we have that  −1 = 1 10 2 8 4 7 5 6 2 9.
�ractically, the whole procedure of �nding the compres� sion can be presented in a table, where the second row refers to the �ow , which includes all the elements necessary for the conditions (a)-(e), while from their application we deduce in the third row the total order of the set (.e last row refers to the �ow  by assigning the numbers of the set [2 to their corresponding blocks of (, according to the following remarks for the blocks   =  1  2 . (i) eir elements have the same ordering (ascending or descending) with those of   .

Conclusions
We have introduced the compression of a meander, directly with the use of blocks of its permutation.e uniqueness of the compression is established by the ordering of the blocks, which is deduced according to its �ow.
Various open questions can arise by the above meanders, when they are used as representatives of large classes of meanders as shown in Table 1.Yet, the main open problem is the reverse procedure of compression.e decompression of a meander to others of larger order having the same traces, number of cuttings, and �ow seems to be the �nal step for integrating the procedures of cutting and compressing meanders, and in parallel being very promising for enumeration results and applications in physical phenomena.