Discrete Mathematics Volume 0, Article ID 575, 6 pages http://dx.doi.org/0.55/0/575 Research Article Compression of Meanders A. Panayotopoulos, and P. Vlamos Department of Informatics, University of Piraeus, Karaoli & Dimitriou 0, 5 Piraeus, Greece Department of Informatics, Ionian University, Plateia Tsirigoti 7, 00 Corfu, Greece Correspondence should be addressed to P. Vlamos; vlamos@ionio.gr Received June 0; Revised August 0; Accepted September 0 Academic Editor: Annalisa De Bonis Copyright 0 A. Panayotopoulos and P. Vlamos. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper refers to the algorithmic transformation of a meander to its uniquely de ned 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.. Introduction A closed meander of order nn is a closed self-avoiding curve crossing an in nite hori ontal line nn times []. 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 [] and they have been studied with nested sets [, ]. More speci cally, in Section, we de ne the ow of a meander consisted by its traces and corresponding blocks. In Section, 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, 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 therestofthepaper[]. A set SS of disjoint pairs of [nnn such that {aaaaaaaaa {aaa aaa a [nnn and for any {aaa aaaa aaaa aaa a aa we never have aaaaaaaaa dd is called nested set of pairs on [nnn. 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 [nnn by NN nn. Two nested sets SS,SS NN nn de ne a permutation σσ on [nnn,suchthatσσσσσσ σ )=jjiff {ii i ii iii i ii and σσσσσσσ σ σσ iff {iii iii i ii, for every ii i iiii. e sets SS,SS are kk-matching if and only if σσ has kk cycles. In the case where kkkk, SS,SS are simply called matching. is de nition is equivalent to the one given in []. 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 aa such thatthereisnopair{ccc ccc c cc with ccccccccccc. Each nested setofpairscontainsatleastoneouterandoneshortpair.. Meanders A meander of order nn is equivalently de ned [] as a cyclic permutation μμμμμ() μμ () μμ(nn) () on [nnn, for which the following properties hold true: μμμμμ μ, and the sets UUU μμ (ii),μμ(iiii) ii i ii ii i i iii i i, LLL μμ (ii),μμ(iiii) ii i ii ii i i iii are both nested and matching. We take all numbers mod nn. Itisclearthatμμμμμμ is odd if and only if ii is odd. In the corresponding geometrical representation, the nested arcs correspond to nested pairs. A pair of nested sets UU, LL should be matching, in order to ()
Discrete Mathematics 0 () () () 7 () 5 6 7 0 (5) 5 6 7 0 5 6 7 6 (7) 5 (6) () (0) () FIGURE : A meander of order. generate a meander. For example, the meander of Figure is of order, with μμ μ μ μμ μμ μ μ μμ μμ μμ μμ μμ μμ μμ μμ 7 6 0 5 0 7 6 5, UUU{{, }, {, 7}, {, }, {, 5}, {6, 7}, {, }, {, 0}, {, 6}, {, }, {5, }, {6, 7}, {, 0}, {, 5}, {, }}, LLL{{, }, {, }, {5, }, {6, }, {7, 0}, {, 7}, {, 6}, {0, 5}, {, }, {, }, {, }, {, }, {5, 6}, {7, }}. e set of all the meanders of order nn is denoted by M nn. Let μμ μ μ nn be a meander crossing a horizontal line. Following [5], for any ii i iiii i ii we consider the vertical line, which shall be called the ii-line, passing through the middle point of the segment (iii ii i ii of the horizontal line. e numbers of those arcs of the meandric curve which are intersected by the ii-line and lie above and beneath the horizontal line of μμ, are called the numbers of cuttings θθθθθθ and θθ (iii, respectively [6]. e sum εεεεεε ε εεεεεε ε εε (iii of the number of those arcs of the meandric curve which are intersected by the ii-line is called the cutting degree of the meander at ii. We notice that θθθθθθ and θθ (iii are of the same parity; hence, εεεεεε is always even [5]. e meandric curve always has points of intersection with the ii-line, which we call traces. Obviously, the number of the traces is equal to εεεεεε. Starting below the horizontal line, we label the traces with the numbers,,,, εεεεεε, knowing that θθθθθθ (resp., θθ (iii) 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 and for ii i ii, we have θθθθθθ θ θ, θθ () = 6, and εεεεεε ε εε. Beginning from trace 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 where the circled elements are the traces. Inthegeneralcase,wehavetheshuffle tt ii =ττ ii () BB ii ττ ii () BB ii ττ ii (εε (ii)) BB ii εεεεεε () with ττ ii () =, ττ ii (),, ττ ii (εεεεεεε being the traces of the meander at ii and BB ii,bb ii,,bb ii εεεεεε the parts of consecutive elements of the meandric permutation μμ, lying between consecutive traces, called blocks of the meander; that is, the block BB ii kk, kk k kkkkkkkk, isthesetoftheconsecutiveelementsof the permutation μμ, which are lying between the traces ττ ii (kkk and ττ ii (kk k kk. ese two traces are called the entrance trace and the exit trace of the block, respectively, while the shuffle tt ii is called o of the meander from the trace () of the ii-line, or for simplicity ii- o. For every kk k kkkkkkkk, ττ ii (kkkkk ii kkττ ii (kk k kk corresponds to the part of the meandric curve starting from the trace ττ ii (kkk and ending at the trace ττ ii (kk k kk, which we denote by cc ii kk. If kk is odd (resp., even), then this curve lies on the le (resp., right) of the ii-line. In Figure,wedenotethecurvecc ii kk by (kkk.. Simple Meanders and Compression Let μμμμ nn and its ow tt ii =ττ ii ()BB ii ττ ii ()BB ii ττ ii (εεεεεεεεε ii εεεεεε. To each block BB ii kk, kk k kkkkkkkk, we correspond a number bb ii kk such that bb ii kk =, if BBii kk is odd,, if BB ii kk is even. (5)
Discrete Mathematics t = 5 0 7 6 5 7 7 6 5 0 6 5 0 7 6 FIGURE 0 () () () (5) 7 () 5 6 7 6 0 5 6 7 (7) 5 (6) () () (0) FIGURE : A meander of order, simple at iiii. e set BBBBBB B BBB ii,bb ii,,bb ii εεεεεε} can be partitioned into three classes BB (iii, BB (iii, and BB (iii, where the set BB (iii consists of the blocks BB ii kk with bb ii kk =and the set BB (iii consists of the blocks BB ii kk with bb ii kk = and θθ (iii i ii ii (kkkk kk ii (kk k kk, while the set BB (iii consists of the blocks BB ii kk with bb ii kk =and ττ ii (kkkk kk ii (kk k kk k kk (iii. A meander μμ μ μ nn is called simple at ii (i.e., simple referring to the ii-line) if BB ii kk, kk k kkkkkkkk. Hence, every block of the sets BB (iii (resp., BB (iii, BB (iii) has exactly one element (resp., two elements). We notice that the following necessary and sufficient condition holds true. A meander is simple at ii if and only if every triple of consecutive terms of its permutation contains elements from both of the sets {,,, iii and {ii i ii ii i ii i i iiii. For example, the meander of Figure is simple at iiii. Wenotethatameandercanbesimpleatmorethanone point. For example, the meander μμμμμμμμμissimple at iiiiand iiii. Let μμμμ nn be a meander that is not simple at nn. Further on, we will study every meander according to its nn-line, so for simplicity we omit the index nn from the notation. Let BBBBBB B {BB,BB,,BB νν } be its already de ned set of blocks, where νν ν νννννν and the blocks BB kk, for kkkkk = {,,, νν ν νν (resp., kkkkk = {,,, ννν), are the ones placed at the le (resp., right) of the nn-line. Each block BB kk BBBBBB lies between the trace ττττττ and the trace τττττ τ ττ ofthe owtt. We denote by γγ kk, kk k kkkkk, the closed interval of [ννν with ends the traces ττττττ, τττττ τ ττ. Given a pair of blocks BB kk,bb λλ BBBBBB, kkkkkkkk or kkk kk k kk, with γγ λλ γγ kk, then the block BB λλ is called internal of the block BB kk,andtheblockbb kk is called external to the block BB λλ.wecaneasilydeducethatthe number of the internal blocks of a block BB kk BBBBBB is equal to (/)( ττττττ τ τττττ τ τττ τ ττ. If we replace the blocks BB kk, kk k kkkkk, of the meander μμ by blocks having one element (resp., two elements) whenever bb kk = (resp., bb kk = ), then we obtain a meander μμ of order nn n nnnnn n kkkkkkkk bb kk (since its blocks have one or two elements, corresponding to crossing points with the horizontal line) and simple at uu u u kkkkk bb kk (counting the points of intersection with the horizontal line to the le of the nn-line). e result of the above replacement is the set of blocks BBBBBB B BBB, BB,,BB νν }, where BB kk = BB uu kk for simplicity. e set BBBBBB is partitioned into the classes BB (uuu, BB (uuu, and BB (uuu corresponding to the classes BB (nnn, BB (nnn, and BB (nnn of the set BBBBBB. When we put a dash upon any existing notation, we refer to the elements of the deduced simple meander μμ. Weeasily obtain that (i) uuu uu areofthesameparity, (ii) εεεεεε ε εεε, θθθθθθ θ θθθθθθ, θθ (uuu u uu (nnn, bb kk =bb kk, (iii) if BB kk BB (nnn (resp. BB kk BB (nnn) and ττττττττττττττττ τ, then its corresponding block BB kk BB (uuu (resp., BB kk BB (uuu) contains one short pair of LL (resp., UU). If BB kk BB (nnn, then its corresponding block BB kk BB (uuu isthepair{ττττττ, τττττ τ τττ. e pair (μμμ μμμ is called central compression or simply compression of the meander μμ. e simple at uu meander μμ
Discrete Mathematics µ = 7 6 5 7 6 0 5 0 7 6 5 7 6 5 7 6 0 5 0 7 6 5 (0) () () (7) 5 7 6 5 7 6 0 5 0 7 6 () (5) (6) () () () FIGURE has as same invariants with the meander μμ the traces and the ow of curves. For example, the compression of the meander μμ of Figure is the pair (μμμ μμμ, where μμ is the meander of Figure and uuuuuuuuuuuuuu.. The Flow tt e traces and the blocks of the ow tt of a meander μμ can be found from its permutation with the help of the subsets UUUUUU U UU and LLLLLL L LL, containing the pairs {μμμμμμμ μμμμμ μ μμμ satisfying the relation min μμ (ii),μμ(iiii) nn n nnn μμ (ii),μμ(iiii). (6) ese pairs have one element belonging to the set {,,, nnn andtheotherbelongingtotheset{nn n nn nn n,, nnn, with UUUUUUU U UUUUUU and LLLLLLL L LL (nnn. According to the absolute value μμμμμμ μ μμμμμ μ μμμ, we place the elements of LLLLLL in decreasing order, while of UUUUUU in increasing order. If UUUUUUU UUUUUU U U, then we correspond the numbers θθ (nnn n nn n n nnn n nn nnn to the ordered pairs of UUUUUU, and the numbers,,, θθ (nnn totheorderedpairsofllllll. If LLLLLL L L (resp., UUUUUU U U),thenwecorrespondtothe ordered pairs of UUUUUU (resp., LLLLLL) the numbers,,, νν. It is obvious that the previous numbers coincide with their corresponding traces. Inordertostartthenn- ow tt from the rst trace, we choose the pair {μμμμμμμ μμμμμμμμμ which corresponds to the trace τττττ τ. If LLLLLL L L, then this pair is the outer pair of LLLLLL with μμμμμμ ) odd and less than μμμμμμ. If LLLLLL L L,thenthisisthepairof UUUUUU with the smallest value of μμμμμμ μ μμμμμ μ μμμ. For example, for the meander of Figure, we have that UU () = {{, }, {, 7}, {, 6}, {, }}, LL () = {{5, }, {6, }, {7, 0}, {, 7}, {, 6}, {0, 5}}. (7) Hence, we choose the pair {5, } of LLLLLL to correspond to the trace τττττ τ τ. Since the permutation μμ iscyclic,wedonotchangethe notation, that is, μμμμμ(zzzz) μμ() μμ () μμ(zz), () which also de nes a partition of [nnn with classes including its consecutive elements, which belong, respectively, to the sets {,,, nnn and {nn n nn nn n nn n n nnnn. ispartitiongivesthe νν classes of the set BBBBBB B BBB,BB,,BB νν }. ractically, at rst we partition the permutation of the meander into classes including the consecutive terms of μμ, which are less or equal to (resp., greater than) nn. us, we have the partition of μμ into blocks, putting at the beginning the last remaining elements and marking the traces. For example, for the meander of Figure we have Figure. 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. e meanders of the compression of the set M nn are partitioned into classes, with elements belonging to the sets M,M,,M nn. In the methods of generating planar permutations [7], we can also include the way to nd the blocks of meanders, their corresponding numbers bb kk, 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 μμμμ nn simple at uu to all possible meanders μμμ M nn, with nnnnn. Table presents the cardinalities of those classes for nnn,,, 0, without taking into account the corresponding uuline. e eros of the rst (resp., second) column verify the fact that if the order nn of the meander is even (resp., odd), then there do not exist meanders of order nn with compression of order (resp., ). We can easily prove that the values of the rst column express that there exist M (nnnnnnn meanders of order nn with compression of order. For meanders of larger order, we should try to calculate the number of different blocks of given orders. 5. Determining the Compression We shall nd the compression of a meander μμ with the help of its nn- ow tt t tttttbb τττττbb τττττττbb νν. We recall that each block BB kk BBBBBB consists of one or two elements. Obviously, the elements of BB kk belong to the sets {,,, uuu (resp., {uu u, uu u uu u u unnn), when kkkkk (resp., kkkkk ). e relative position of these points de nes a relation of preceding for the blocks of the set BBBBBB; hence, the block BB pp precedes the block BB qq (BB pp BB qq ), iff min BB pp < min BB qq.
Discrete Mathematics 5 TABLE : epartitionofthesetm nn into the classes M nn. nnnnnn 6 0 6 0 Total 0 6 0 0 6 0 6 0 0 6 0 66 0 6 7 76 0 60 760 7 0 6 0 000 7 500 0 0 0 05 66 0 070 500 60 66 66 5 0 5 0 0 57 6 6 060 557 77 0 6 0 560 k 5 6 7 0 t 5 0 7 6 5 7 7 6 0 5 6 5 0 7 6 r 0 7 5 6 t 7 6 0 7 5 5 6 0 7 5 6 FIGURE 5 is relation is de ned by the following conditions. (a) If BB pp BB (nnn, BB qq BBBBBB, with γγ qq γγ pp and pppppppp (resp., pppppppp ), then BB pp BB qq (resp., BB qq BB pp ), since the block BB qq is internal of the block BB pp. (b) If BB pp BB (nnn n nn (nnn, BB qq BB (nnn, with γγ qq γγ pp = and ppp pp p pp (resp., pppppppp ), then BB pp BB qq (resp., BB qq BB pp ), since this is imposed by the nature of the meander. (c) If BB pp,bb qq BB (nnn n nn (nnn, with γγ qq γγ pp and ppp pp p II or ppp pp p pp, then BB pp BB qq, since the block BB qq is internal of the block BB pp. (d) If BB pp,bb qq BB (nnn, with γγ qq γγ pp =, min γγ qq < min γγ pp and ppp pp p pp (resp., pppppppp ), then BB pp BB qq (resp., BB qq BB pp ), since this is imposed by the nature of the meander. e case where BB pp,bb qq BB (nnn is similar. Obviously, the above results do not cover the cases of two blocks, the one belonging to the set BB (nnn, and the other to the set BB (nnn, where none of them is internal to the other. In order to obtain a unique solution, we havetomakethefollowingchoice. (e) If BB pp BB (nnn, BB qq BB (nnn, with min BB pp < min BB qq and ppp pp p pp or pppppppp, then BB pp BB qq (resp., BB qq BB pp ), where BB kk =bb kkk bb kkkk, bb kkk is the rst element of BB kk, bb kkkk isthelastelementofbb kk and ωω ω ωωω kk. From the above conditions (a) (e), we obtain an ordering for the blocks of the set BBBBBB, concerning their relation of preceding, that is, BB rrrrr, BB rrrrr,,bb rrrrrrr. For example, applying the above conditions for the meander of Figure,weobtainthatBB BB BB BB 5 BB 7 and BB BB 6 BB BB 0 BB. Indeed, BB BB, BB 5, BB 7, BB due to the condition (a), BB BB 5 due to the condition (b), BB BB due to the condition (e), BB 5 BB 7 due to the condition (a), and nally BB BB 5 due to the condition (b). Similarly, we obtain the ordering for the rest of the blocks. Hence, rrrrrrrrrrrrrr. In the general case, the ordering is deduced from a Hamiltonian path of the directed graphs with vertices the elements of the set II (resp., II ) and arcs the pairs (ppp ppp p pp (resp., (ppp ppp p pp )suchthatbb pp BB qq. WenotethatBB pp BB qq iff rr (ppp p pp (qqq,giventhatrr (kkk de nes the position of the block BB kk atthetotalorderofbbbbbb. Forourexample,wehavethatrr =0756. ractically, the whole procedure of nding the compres sion can be presented in a table, where the second row refers to the ow tt, 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 BBBBBB. e last row refers to the ow tt by assigning the numbers of the set [nnn to their corresponding blocks of BBBBBB, according to the following remarks for the blocks BB kk = bb kkk bb kkk. (i) eir elements have the same ordering (ascending or descending) with those of BB kk. (ii) When their elements are not consecutive, then bb kkk bb kkk = ττττττ τ τττττ τ τττ. For example, for the meander of Figure, we have Figure 5. Hence, μμμμμμμμμμμμμμμμμμμμμμμμμμ, with uuuuuuuuuuuuuu.
6 Discrete Mathematics 6. 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. 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. References [] S. K. Lando and A. K. Zvonkin, Plane and projective meanders, eoretical Computer Science, vol. 7, no. -, pp. 7,. [] P. osenstiehl, Planar Permutations de ned by two interesting Jordan curves, in Graph eory and Combinatorics, pp. 5 7, Academic Press, New York, NY, USA,. [] J. Barraud, A. Panayotopoulos, and P. Tsikouras, Properties of closed meanders, Ars Combinatoria, vol. 67, pp. 7, 00. [] M. A. La Croix, Approaches to the enumerative theory of meanders [M.S. thesis], University of Waterloo, 00. [5] A. Panayotopoulos and P. Vlamos, Cutting degree of meanders, in Proceedings of the st Mining Humanistic Data Workshop, 0. [6] A. Panayotopoulos and P. Vlamos, Meandric polygons, Ars Combinatoria, vol. 7, pp. 7 5, 00. [7] A. Panayotopoulos, Generating planar permutations, Journal of Information and Optimization Sciences, vol., no., pp. 7, 7.
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