ON THE REPRESENTATION OF MEANDERS

Transcription

1 ON THE REPRESENTATION OF MEANDERS REINHARD O. W. FRANZ ABSTRACT. We will introduce a new approach for studying plane meanders. The set of all meanders of order n possesses a natural order structure and forms a graded poset. We will show how these representations can be used to develop an efficient and very flexible construction algorithm and how to obtain counting formulae for meanders. 1. INTRODUCTION Probably the oldest version of the meander problem appears to be the stamp folding problem, which asks for the number of ways in which a strip of n stamps can be folded. This question seems to be first mentioned by Edouard Lucas in 1891 (cf. [25]) and attracted some interest in the first half of the past century. However, none of the ideas pursued led to a solution of the problem. In the seventies, the problem regained some new interest, and new ideas were introduced, but still no solution was presented (cf. [20], [26], [27]). About twenty years later, V. Arnol d (cf. [3]) brought new attention to the problem in connection with differential geometry relating it with Hilbert s 16th problem concerning the number of ovals of an algebraic curve in the plane. The more general problem considered now can be stated as follows: Count the number of possible configurations of a closed non-selfintersecting ringroad crossing a given river at a given number of bridges. As we will see in Section 2, stamp foldings can be identified with certain types of simple meanders, i.e. meanders with only two ends, and are therefore also accessible to the methods introduced in this note. Although the meander problem can be stated in very simple terms, it has proven to be unexpectedly difficult and still awaits a solution. In the meantime, it has become apparent that the meander problem emerges in a variety of different fields, ranging from mathematics to computer science to physics. Besides its application in differential geometry mentioned before, the problem also appears in the classification of 3-manifolds (cf. Date: May 21, Mathematics Subject Classification. Primary: 05A15, 05A18 Secondary: 06A07, 06A08. Key words and phrases. Meanders, Noncrossing Partitions, Partitions. 1

2 2 REINHARD O. W. FRANZ [21]). In computer science, the meander problem arises in the context of sorting Jordan sequences (cf. [17]). Also physicists became interested in the meander problem in connection with the so-called critical phenomena, Meanders can be considered as a certain type of self-avoiding walks. (cf. [7]). In this note, we present a new approach for studying plane meanders by establishing bijections between the set of meanders and certain sets of pairs of noncrossing partitions, pairs of permutations, and the set of meandric trees. Using the representation of meanders by certain pairs of partitions, we show that the set of meanders possesses a natural order structure and forms a graded poset with respect to this order (cf. [9]). In addition, we show that this order structure can be used to obtain numerical results. 2. PRELIMINARIES Let N denote the set of positive integers 1, 2,..., and N m, for m N, the set of positive integers less than or equal to m. Moreover, let E := R 2 denote the Euclidean plane and E + and E the upper and lower half planes {(x, y) E y 0} and {(x, y) E y 0} of E, respectively. For a given positive integer n N, consider a fixed oriented line L in the Euclidean plane E := R 2 and 2n distinct points p 1,p 2,...,p 2n on L. In order to keep the notation simple, we will assume that L = {(x, 0) x R} and p i =(i, 0) for i =1, 2,...,2n. A system of meandric curves of order n is defined to be a non-selfintersecting (not necessarily connected) closed curve C E, which transversally intersects the line L at precisely the points p 1,...,p 2n. Two systems of meandric curves C 1 and C 2 are said to be equivalent if they can be deformed into each other by an isotopy of E that fixes the points of the line L. The equivalence class M =[C] of a system of meandric curves C in E is called a system of meanders of order n, or simply a meander of order n if C is connected. We denote the set of all systems of meanders of order n by S n and the subset of all meanders of order n by M n. For a system of meanders M S n let π 0 (M) denote the set of connected components of M. If M is a system of meanders of order n and C M then we call the path component S π 0 (E ± \ C) an end of M if there exists precisely one i N n such that ]p 2i 1,p 2i [ S, where ]p, q[ denotes the open line segment between the points p, q E. By the definition of equivalence of meandric curves this definition is independent of the representative C of M. A system of meanders M is defined to be simple if it only possesses 2k ends, where k =#π 0 (M) denotes the number of connected components of M.

3 ON THE REPRESENTATION OF MEANDERS 3 FIGURE 1. The simple meander associated with an open meander By modifying the definition of systems of meanders above, allowing the number of points p 1,...,p m on L be an arbitrary positive integer m N and replacing the closed curve C E by an open curve, we arrive at the notion of a system of open meanders of order m or open meanders of order m if C is connected. Open meanders can be considered as a subclass of (closed) meanders: Proposition 1. There exists a natural bijection between the set of all systems of open meanders of order n and the set of all systems of simple meanders of order 2n. Proof. The proof is straightforward. For the construction of the bijection consult Figure THE ASSOCIATED PAIR OF PARTITIONS The most natural of the three representations of meanders introduced in this note appears to be the representation in terms of partitions, which was introduced in [9] and which we will outline in this section. For a more extensive discussion of this representation consult [9].

4 4 REINHARD O. W. FRANZ FIGURE 2. The system of meanders M = M P given by the pair of partitions P = (1, 5, 6/2/3, 4/7, 8/9; 1, 2, 3/4, 5, 8, 9/6, 7) L Clearly, each system of meanders M S n of order n defines a pair (P +,P ) of partitions of the set N n. This can be obtained by first choosing an arbitrary system of meandric curves C in M and then assigning to each connected component S π 0 (E ± \ C) the set A S := {i N n ](2i 1, 0), (2i, 0)[ S} N n, It is easy to see that A S only depends on S and is independent of the choice of C since equivalence of meandric curves is defined in terms of isotopy relative to the line L. Therefore, the sets P ± := P ± (M) :={A S S π 0 (E ± \ C)} constitute a well-defined topological invariant P n (M) =(P +,P ) of the isotopy class M. In fact, it follows immediately from the definition of meanders that both P + and P form noncrossing partitions of the set N n. Recall that a partition P of N n is noncrossing if, whenever a<b<c<d for a, b, c, d N n and a, c A and b, d B for some A, B P, then in fact A = B. The set of all noncrossing partitions of N n will be denoted by NC n. We will follow the common practice and write for a partition P of N n simply P = B 1 /B 2 /.../B m, where the blocks B i (i =1,...,m)ofP are listed in ascending order of their minimum element. Conversely, to any pair P =(P +,P ) of noncrossing partitions of N n we can assign a system of meanders M P of order n called the topological realization of P, such that P n (M P )=P.

5 ON THE REPRESENTATION OF MEANDERS 5 (Cf. Figure 2.) To this end first observe that each partition P of N n defines a unique permutation σ P of N n whose cycles coincide with the classes of P arranged in ascending order. Then, denoting for i 1,i 2 N 2n the arc connecting the points p i1 and p i2 in the halfplane E ± by C ± (i 1,i 2 ), the system of meandric curves C P defining M P is given by ( C P = C ε 2j, 2σε (j) 1 ) j N n,ε {±} In fact, it can be shown that the association M P n (M) defines a bijection P n : Proposition 2. The association M P n (M) defines for any n N a bijection P n from the set S n of all systems of meanders of order n onto the set NC n NC n of all pairs of noncrossing partitions of N n, whose inverse is given by assigning to each pair of noncrossing partitions P of N n the system of meanders M P as defined above. Proof. The proof is straightforward. We call P n (M) the pair of noncrossing partitions associated with the system of meanders M and call a pair of noncrossing partitions P of N n meandric if P is an element of the image P n (M n ) of M n. We will now characterize the subset of all meandric pairs of noncrossing partitions. To this end, consider the finite bipartite graph Γ P := (V,E; I) which can be associated to any pair of partitions P =(P +,P ) of N n by taking as vertex set V the disjoint union of P + and P, as set of edges E the set N n and as incidence relation the set I := {(C, j) V E j C}. A pair of partitions P =(P +,P ) is said to be connected if the associated graph Γ P is connected, which obviously is equivalent to the statement that whenever the unions of two nonempty subsets P + P + and P P coincide, then P + = P + and P = P. We call each pair (P +,P ) of subsets P + P + and P P for which X = Y, X P + Y P a connected component of P and denote the set of all connected components of P by π 0 (P). Clearly, by definition, we have #π 0 (P) =#π 0 (Γ P ), where π 0 (Γ P ) denotes the set of connected components of the graph Γ P associated with P. Now, given an arbitrary system M of meanders of order

6 6 REINHARD O. W. FRANZ n, choose any C M and consider the set B C := {O π 0 (E \ C) ](2i 1, 0), (2i, 0)[ O for some i N n }. Then it can be easily shown that #B C =#π 0 (Γ P ) and #π 0 (M) =#π 0 (C) = #π 0 ( O), O B C where O denotes the boundary of the set O E with respect to the standard topology of E. Moreover, with c Γ denoting the number of independent cycles of a given graph Γ, we can show that for any O B C we have #π 0 ( O) =c Γ(O) +1, where Γ(O) is the unique component of Γ P for which ](2i 1, 0), (2i, 0)[ O for all i E(O), and where E(O) denotes the set of edges of Γ(O). Hence, #π 0 (M) = ( cγ(o) +1 ) = c ΓP +#B C, O B C and thus, (1) #π 0 (M) =c ΓP +#π 0 (Γ P ). It is well known that the number of independent cycles c Γ of a graph Γ= (V,E; I) is given by the identity (2) c Γ =#E #V +#π 0 (Γ), from which, using (1), we obtain the identity (3) #π 0 (M) =2#π 0 (Γ P )+n #P + #P, which allows us to compute the number of connected components of a system of meanders M in terms of the associated pair of noncrossing partitions. Meandric pairs of noncrossing partitions can now be easily characterized: Theorem 1. For a pair P = (P +,P ) of noncrossing partitions of N n (n N), the following statements are equivalent: (1) P is meandric. (2) #P + +#P = n +1and P is connected, i.e. the associated graph Γ P is a tree.

7 ON THE REPRESENTATION OF MEANDERS 7 Proof. Suppose P =(P +,P )=P n (M) for some meander M in M n. Then by definition #π 0 (M) = 1, and by (1) we obtain c ΓP = 0 and #π 0 (Γ P )=1, from which by (3) follows that #P + +#P = n +1. Conversely, if #π 0 (Γ P )=1and #P + +#P = n +1then (3) implies that the associated system of meanders M = P 1 (P) has precisely #π 0 (M) =2 1+n (n +1)=1connected components. Therefore, M is a meander (of order n) and thus P meandric. It is well-known that the number of noncrossing partitions of N n coincides with the nth Catalan number (cf. [6]) Hence the bijection between the sets S n and NC n NC n established above yields another proof for the known fact that the number of all systems of meanders of order n coincides with (C n ) 2 (cf. [23]). 4. THE ASSOCIATED PAIR OF PERMUTATIONS In this section we present a construction that leads to a bijection between systems of meanders and certain pairs of permutations. In the case of plane meanders, on which we focus in this note, the pair of permutations associated with a given system of meanders is just the ordered version of the associated pair of partitions introduced in the previous section. However, the construction presented below extends to a bijection between systems of meanders of arbitrary genus and pairs of permutations (see [11]), while the former construction only yields an injection. For a given n N let S 2n denote the subset of all permutations of S 2n which map even numbers on odd and odd numbers on even. For each such permutation σ S 2n, we define two permutations σ + and σ of S n, which we will call the positive and negative components of σ, respectively, by setting σ + (i) := 1 ( ) (4) σ 1 (2i)+1 2 and (5) σ (i) := 1 2 ( ) σ(2i)+1 for all i N n. Conversely, we define for each pair of permutations (τ 1,τ 2 ) S n S n a permutation ρ = ρ τ1,τ 2 in S 2n by setting { ( 2τ 1 i+1 ) 1 (6) ρ τ1,τ 2 (i) := 2, if i is odd ( 2τ i 2 2) 1, else. for all i N 2n. These two operations are inverse to each other:

8 8 REINHARD O. W. FRANZ Proposition 3. (1) The assignment σ (σ +,σ ) defines for any n N n a bijection Ψ n : S 2n S n S n from S 2n onto S n S n, whose inverse is given by (τ 1,τ 2 ) ρ τ1,τ 2. (2) Suppose σ S 2n is given and σ +,σ S n are its positive and negative components, respectively. Then σ 2 (2i 1) = 2 ( ) σ σ+ 1 (i) 1 σ 2 (2i) =2 ( σ+ 1 σ ) (i) for all i N n. Proof. (1) We have to show that (i) ρ σ+,σ = σ for any σ S 2n and (ii) (ρ τ1,τ 2 ) + = τ 1 and (ρ τ1,τ 2 ) = τ 2 for all τ 1,τ 2 S n. (i) Given an arbitrary permutation σ in S 2n we first observe that σ +(i) = 1 2 (σ 1 (2i)+1)is equivalent to σ+ 1 (i) = 1σ(2i 1) for any i N 2 n. Thus, { ( 2σ 1 i+1 ) + ρ σ+,σ (i) = 2, if i is odd ( 2σ i 2) 1, else. { 2 1 = σ( 2 i+1 1 ), if i is odd 2( ( 2 ) σ 2 i 2) +1 1, else. = σ(i) for all i N 2n. (ii) Now let τ 1,τ 2 S n be given. By definition, the permutation ρ τ1,τ 2 maps odd numbers on even and even numbers on odd. Equation (6) implies that ρ 1 τ 1,τ 2 (2i) =2τ 1 (i) 1 for all i N n. Therefore (ρ τ1,τ 2 ) + (i) = 1 ( ρ 1 τ 2 1,τ 2 (2i)+1 ) = 1 2( 2τ1 (i) 1+1 ) = τ 1 (i), and (ρ τ1,τ 2 ) (i) = 1 ( ρτ1,τ 2 2 (2i)+1 ) = 1 2( 2τ2 (i) 1+1 ) = τ 2 (i) for all i N n. (2) The identities follow directly from equation (6): σ 2 (2i 1) = σ ( 2σ+ 1 (i) ) ( = 2σ σ 1 + (i) ) 1, σ 2 (2i) = σ ( 2σ (i) 1 ) ( = 2σ+ 1 σ (i) ) for all i N n.

9 ON THE REPRESENTATION OF MEANDERS 9 FIGURE 3. The permutation σ M corresponding to the meander M can be obtained from M by traversing its curve components in the direction given by the arrows as shown below always starting at the vertex with the smallest number. The seqences of vertices crossed by traversing each component constitute the cycles of σ M. The permutation associated with the meander M below is: σ M = (1, 10, 7, 6, 3, 2, 13, 14)(4, 5, 8, 9)(11, 16, 15, 12). M L To each system of meanders M of order n, there can be assigned a unique permutation σ M (cf. Figure 3) of N 2n whose cycles consist of the sequence of indices of those points p ij crossed by traversing the individual curve components of an arbitrary system of meandric curves of M moving from odd indices to even on top and from even to odd on the bottom. σ M maps even numbers onto odd and odd numbers onto even and is therefore an element of S 2n. Clearly this assignment establishes an injective map Q n from S n into the symmetric group S 2n, which obviously is not surjective. σ M is called the permutation associated with the system of meanders M. We define a permutation σ to be meandric if σ Q n (M n ). A permutation σ S n is defined to be ascending if all its cycles can be arrayed in ascending order by a cyclic permutation of their elements. σ is said to be noncrossing if the set N n / σ of all orbits of the subgroup σ S n in N n is a noncrossing partition of N n. A permutation σ S 2n is called partially noncrossing if the derived permutations (7) σ (+) := n ( ) 2i 1,σ(2i 1) i=1

10 10 REINHARD O. W. FRANZ and (8) σ ( ) := n ( ) 2i, σ(2i) i=1 are noncrossing. Clearly, the permutation σ M associated with a system of meanders M of order n is partially noncrossing. Moreover, a permutation σ S 2n is the permutation associated with a system of meanders of order n if and only if σ S 2n and σ is partially noncrossing. We will now characterize meandric permutations in terms of their positive and negative components. The following lemmas establish some partial results. Lemma 1. For any n N, i 0 N n and σ Q n (S n ) the following statements are equivalent: (1) σ ± (i 0 ) <i 0 ; (2) i 0 =max{σ± k (i 0) k N n } and σ ± (i 0 )=min{σ± k (i 0) k N n } Proof. The proofs are similar for σ + and σ. Therefore, we will verify the assertion only for σ +. (1 2) Let O := {σ+ k (i 0) k N} and suppose that O\[σ + (i 0 ),i 0 ]. Then there exist distinct elements i 1,i 2 O such that i 1 [σ + (i 0 ),i 0 ), i 2 / [σ + (i 0 ),i 0 ] and σ + (i 1 )=i 2. But this contradicts the assumption that σ is partially noncrossing. In fact, in the case that i 2 <σ + (i 0 ),wehave σ + (i 1 )=i 2 <σ + (i 0 ) i 1 <i 0, which, by equation (4), is equivalent to σ 1 (2i 1 ) = 2i 2 1 < σ 1 (2i 0 ) 2i 1 1 < 2i 0 1, and implies σ 1 (2i 1 ) <σ 1 (2i 0 ) < 2i 1 < 2i 0. Hence, σ (+) is crossing, in contradiction to σ Q(S n ). In the remaining case, where i 2 >i 0,wehaveσ + (i 0 ) i 1 < i 0 < i 2 = σ + (i 1 ), which is equivalent to σ 1 (2i 0 ) 2i 1 1 < 2i 0 1 < 2i 2 1=σ 1 (2i 1 ) and implies σ 1 (2i 0 ) < 2i 1 < 2i 0 σ 1 (2i 1 ). Considering, that σ S 2n (i.e. σ maps even numbers onto odd, and vice versa!), we obtain σ 1 (2i 0 ) < 2i 1 < 2i 0 <σ 1 (2i 1 ), which also contradicts the assumption that σ (+) is noncrossing. (2 1) Trivial. The positive and negative components of a permutation associated with a system of meanders are noncrossing: Lemma 2. Suppose n N and σ Q n (S n ). Then σ + and σ are noncrossing. Proof. We will show the contrapositive of the assertion and can again confine ourselves to the case +. Clearly, a permutation is crossing if and only if its inverse is crossing. Moverover, any crossing of σ+ 1 induces a crossing

11 ON THE REPRESENTATION OF MEANDERS 11 of σ (+). In fact, applying the equation (9) σ+ 1 (i) = 1 σ(2i 1), 2 which is clearly equivalent to equation (4), to any crossing sequence i 1 <i 2 <σ 1 + (i 1) <σ 1 + (i 2) for σ 1 +, yields the corresponding crossing sequence 2i 1 1 < 2i 2 1 <σ(2i 1 1) <σ(2i 2 1) for σ (+). This remains true if we permute i 1 and σ+ 1 (i 1 ) and/or i 2 and σ+ 1 (i 2 ), respectively. Note that the converse of the assertion stated in Lemma 2 is generally not true. Consider the following: Example 1. The permutation σ =(1, 6, 3, 4, 5, 10, 9, 14, 13, 12, 7, 8, 11, 2) in S 14 has the positive component σ + =(1, 6, 7, 5, 3)(2)(4), which clearly is not crossing. However, the permutation σ (+) =(1, 6)(3, 4)(5, 10)(7, 8)(9, 14)(11, 2)(13, 12) is crossing, as can be easily seen. The following lemma states the conditions for σ ±, which ensure that σ (±) is noncrossing: Lemma 3. Let n N and σ S 2n and suppose that the permutation σ ± is noncrossing and ascending. Then the permution σ (±) is noncrossing. Proof. We prove the contrapositive of the implication above and confine ourselves again to verifying the asserting only for σ + and σ (+). Suppose σ (+) is crossing. Then. there exist postive integers i 1,i 2 N n such that or 2i 1 1 < 2i 2 1 <σ(2i 1 1) <σ(2i 2 1), 2i 1 1 <σ(2i 2 1) <σ(2i 1 1) < 2i 2 1, σ(2i 1 1) < 2i 2 1 < 2i 1 1 <σ(2i 2 1) σ(2i 1 1) <σ(2i 2 1) < 2i 1 1 < 2i 2 1.

12 12 REINHARD O. W. FRANZ But using equation (4) each of the crossing sequences for σ induces a corresponding crossing sequence for σ+ 1 : i 1 <i 2 σ+ 1 (i 1 ) <σ+ 1 (i 2 ), i 1 σ+ 1 (i 2) <σ+ 1 (i 1) <i 2, σ+ 1 (i 1) <i 2 <i 1 σ+ 1 (i 2) or σ+ 1 (i 1) <σ+ 1 (i 2) <i 1 <i 2. If σ + is noncrossing then the elements i 1,i 2,σ+ 1 (i 1 ) and σ+ 1 (i 2 ) must belong to the same orbit of σ +, which clearly, in none of the four cases listed above, is ascending. Hence σ + is not ascending. We can now characterize the permutations σ of Q n (S n ) and Q n (M n ) in terms of their positive and negative components: Theorem 2. Let n N (n >0), and σ S 2n. Then σ Q n(s n ) if and only if σ + and σ are noncrossing and ascending. Furthermore, σ is meandric if and only if in addition σ+ 1 σ is cyclic (i.e. σ+ 1 σ is an n cycle). Proof. ( ) Suppose σ Q n (S n ). Then the permutations σ + and σ are ascending by Lemma 1 and noncrossing by Lemma 2. ( ) Now suppose that σ + and σ are ascending and noncrossing. Then by Lemma 3 the permutations σ (+) and σ ( ) are noncrossing, i.e. σ is partially noncrossing. Hence, σ Q n (S n ). The additional statement follows immediately from part (2) of Proposition THE ASSOCIATED GRAPH We showed in Section 3 that we can associate with each system of meanders M a well-defined finite bipartite graph Γ P, where P := P M := P n (M) denotes the unique pair of noncrossing partitions associated with M. This association is clearly not one-to-one as the following examples indicates: Example 2. Let M 1 and M 2 denote the meanders defined by the pairs of noncrossing partitions P 1 := (1, 2/3; 1, 3/2) and P 2 := (1, 3/2; 1, 2/3). Clearly M 1 and M 2 are distinct, but their associated graphs Γ i := (V i,e i ; I i ) (i =1, 2) coincide, since V 1 = V 2 = {{1, 2}, {1, 3}, {2}, {3}}, E 1 = E 2 = {1, 2, 3}, and I 1 = I 2 = {(v, j) V 1 E 1 = V 2 E 2 j v}. However, we can turn this association into a 1-1-correspondence by introducing a suitable order for the edges of Γ P.

13 ON THE REPRESENTATION OF MEANDERS 13 Recall that a graph Γ=(V := V + V,E; I) is said to be bipartite if V + V = and for any e E the set Γ(e) :={v V (v, e) I} of vertices incident with the edge e contains precisely one element of V + and of V, which is equivalent to the condition that the graph Γ does not possess any odd cycles. With any graph Γ=(V,E; I) and pair of subsets V +,V V of its vertex set V, we can assign a pair of subsets E Γ =(E +,E ) of the power set of E defined by E ± := {Γ(v) v V ± }, where Γ(v) :={e E (v, e) I} for all v V. Then, we can characterize bipartite graphs in terms of E Γ : Lemma 4. Suppose Γ=(V,E; I) is a graph and V = V + V a partition of V. Then Γ=(V +,V,E; I) is bipartite if and only if E Γ =(E +,E ) is a pair of partitions of E. Proof. If Γ is bipartite, then for any e E there exists a unique v + V + and v V such that Γ(e) ={v +,v }. It follows that e Γ(v + ) and e Γ(v ) und thus Γ(v) = Γ(v) =E. v V + v V Moreover, since V + V =, Γ(v 1 ) Γ(v 2 )= for all v 1,v 2 that are both in V + or both in V. Hence, E is a partition of E. Conversely, if E + and E are partitions of E, for any e E there exists a unique v + V + and v V such that e Γ(v + ) and e Γ(v ), which implies Γ(e) ={v +,v }. Hence Γ is bipartite. We will now consider systems (V +,V,E; I; σ) consisting of a finite bipartite graph (V := V + V,E; I) together with a bijection σ : E N n, where n = #E. Two such systems Γ = (V +,V,E; I; σ) and Γ = (V +,V,E ; I ; σ ) are defined to be isomorphic if there exist bijections ν : V + V V + V and η : E E such that ν(v + )=V +, ν(v )=V, (ν η)(i) =I, and σ η = σ. The isomorphism class G =[Γ]of each such system Γ=(V +,V,E; I; σ) is called a marked graph. We denote the set of all marked graphs with precisely n edges by G n. Each marked graph G in G n determines a unique pair of partitions P G = (P +,P ) of N n that can be obtained by first selecting an arbitrary representative Γ=(V +,V,E; I; σ) G and then defining P ± := {σ ( Γ(v) ) v V ± }.

14 14 REINHARD O. W. FRANZ Clearly, P G is independent of the choice of Γ G. Indeed, if Γ=(V +,V,E; I; σ) and Γ =(V +,V,E ; I ; σ ) are elements of G and ν and η the bijections establishing the isomorphism from Γ onto Γ, then we have for all e E: σ(e) σ ( Γ(v) ) (v, e) I (ν(v),η(e)) I, which in turn is equivalent to σ(e) =σ ( η(e) ) σ ( Γ ( ν(v) )). Hence, σ ( Γ(v) ) = σ ( Γ (ν(v)) ) for all v V := V + V, and thus, since ν(v ± )=V ±, {σ ( Γ(V ) ) v V ± } = {σ ( Γ (v ) ) v V ± }. Therefore, we obtain: Lemma 5. The association G P G establishes a well-defined bijection P n : G n Π n Π n from the set all marked graphs with n edges into the set of all pairs of partitions of N n. Its inverse H n :Π n Π n G n is given by P =(P +,P ) [(P +,P, N n ; I P ;id n )], where I P := {(X, j) (P + P ) N n j X}. Proof. Indeed, if G G n and Γ = (V +,V,E; I; σ) G and P = (P +,P )=P n(g) with P ± = {σ(γ(v)) v V ± }, then H n (P n(g)) = [(P +,P, N n ; I P ;id n )]. Clearly, Γ and Γ P := (P +,P, N n ; I P ;id n ) are isomorphic. In order to verify this, note that the maps ν : V ± P ± : v σ(γ(v)) and η = σ : E N n are bijective. Moreover, for any v V + V and e E holds (v, e) I σ(e) σ(γ(v)) (σ(γ(v)),σ(e)) I P, and id n σ = σ. Conversely, if P =(P +,P ) Π n Π n, then P n (H n(p)) = P n (Γ P)= P := (P +,P ), where Γ P =(P +,P, N n ; I P ;id n ). We will show that P = P. To this end, observe that Γ P (v) ={i N n i v} = v for v V ± = P ±. This implies P ± = {id n(γ(v)) v P ± } = {v v P ± } = P ±, and thus P = P. The maps P n and P n induce an injection from S n into G n assigning to each system of meanders M a well-defined marked graph G M. Clearly, the pairs of partitions associated with M and G M coincide:

15 ON THE REPRESENTATION OF MEANDERS 15 Proposition 4. For any n N let the maps P n : S n NCn NC n and P n : G n Πn Π n be defined as above. Then there exists a unique injective map G n : S n G n which makes the following diagram commutative: S n P n Π n Π n G n P n G n Proof. The assertion follows immediately from the assumptions that P n is bijective and P n injective. The map G n =(P n ) 1 P n assigns to each system of meanders M S n the marked graph G M := [(P +,P, N n ; I;id n )], where P =(P +,P ) is the pair of noncrossing partitions associated with M, and I := {(X, j) (P + P ) N n j X}. Clearly, G n satisfies the identity P n G n = P n. We call G M = G n (M) the marked graph associated with the system of meanders M. A marked graph G G n is said to be meandric if G G n (M n ). The subsets G n (S n ) and G n (M n ) can now be easily characterized: Theorem 3. Suppose G G n is a marked graph with n edges. Then G G n (S n ) if and only if P G is noncrossing. Furthermore, G is meandric if and only if P G is noncrossing and G is a tree. Proof. Suppose G G n (S n ). Then there exists a system of meanders M S n such that G n (M) =G. Since the pair of partitions P M is noncrossing by Theorem 1, and coincides with P G by Proposition 4, P G is also noncrossing. Now suppose, conversely, that P G is noncrossing. Then, using the fact that im(p n )=NC n NC n, there exists a system of meanders M S n such that P n (M) =P G. Since P n(g) =P G and, by Proposition 4, also P n (G n(m)) = P G, using the injectivity of P n, it follows immediately that G n (M) =G, i.e. G G n (S n ). The second assertion follows immediately from Theorem 1. Remark 1. Theorem 3 has a simple geometric interpretation: The marked graph G =[(V +,V,E; I; σ)] G n is the graph of a system of meanders M S n (n N) if and only if the two following conditions are satisfied: (1) The embedding of G, given by arranging the edges of E around each vertex of V + and V counterclockwise and clockwise, respectively, with regard to the order of E induced by the bijection σ, is planar.

16 16 REINHARD O. W. FRANZ FIGURE 4. The marked graph G = G M corresponding to the system of meanders M given by the pair of partitions P =(1, 5, 6/2/3, 4/7, 8/9; 1, 2, 3/4, 5, 8, 9/6, 7) (2) The path C E obtained by cyclically connecting the midpoints of the edges of E in the order given by σ, without crossing (the embedding of) G at any other point, does not intersect itself. Note that C separates the vertices of V + from those of V. (Cf. Figure 4 and Figure 2). 6. A PARTIAL ORDER ON M n For a more extensive presentation of this material we refer to [9], [10]. Recall that the set Π n of all partitions of N n can be ordered by refinement, under which two partitions P and P of N n satisfy P P if every class (block) of P is contained in some class of P. It is well known that Π n forms a lattice under refinement ordering [1], [32] and that its restriction to the subset of noncrossing partitions is also a lattice [22], [5], [30]. By taking the cross product of the poset NC n with its dual poset NC n, we obtain a poset structure on NC n NC n, where two pairs of noncrossing partitions (P, Q) and (P,Q ) satisfy (P, Q) (P,Q ) if and only if P P and Q Q. This induces a partial order on the set N n of all meandric pairs of noncrossing partitions. Note that N n is not a lattice, since for example (1/2/3, 4; 1, 4/2, 3) and (1/2/3, 4; 1, 2, 4/3) have (1, 2/3, 4; 1, 4/2/3) and (1/2, 3, 4; 1, 4/2/3) as minimal (noncomparable) upper bounds. However, N n is bounded by ˆ0 :=(1/2/ /n;1, 2,...,n)

17 ON THE REPRESENTATION OF MEANDERS 17 FIGURE 5. The Hasse diagram of (M 3, ). 1, 2, 3 1/2/3 1, 2/3 1, 3/2 1, 2/3 1/2, 3 1, 3/2 1, 2/3 1, 3/2 1/2, 3 1/2, 3 1, 2/3 1/2, 3 1, 3/2 1/2/3 1, 2, 3 and ˆ1 :=(1, 2,...,n;1/2/ /n) as minimum and maximum elements, respectively. N n is pure, since all its maximal chains have the same length, namely n, which implies that all unrefinable chains between two comparable elements have the same length. As a finite, bounded and pure poset, N n is a graded poset, and therefore possesses a well-defined rank function ρ n which assigns to each element (P, Q) N n the common length, namely #Q 1, of all unrefinable chains from ˆ0 to (P, Q). This implies that N n has height n 1 (cf. Figure 5). Clearly, the bijection P n : M n N n induces an isomorphic order structure on the set M n of all meanders of order n, under which two meanders M and M satisfy M M if and only if P n (M) P n (M ). We summarize: Theorem 4. For any n N the set M n of all meanders of order n forms a graded poset of height n 1 under the order induced by P n : M n N n. 7. APPLICATIONS: CONSTRUCTING AND ENUMERATING MEANDERS We will now outline how the representations of meanders introduced above can be used for the construction and enumeration of meanders. We refer the reader interested in the details to [12] and [14]. With regard to the results from Section 3 and Section 5 meanders can be considered as a certain kind of ordered trees. Utilizing this interpretation

18 18 REINHARD O. W. FRANZ of meanders we will first introduce a map R n : N n N n which assigns to each meander of order n and rank k<n 1acertain meander of the same order but rank k +1. This map, which we will call the reduction map for N n, since it reduces the number of internal nodes of the associated meandric tree, is a very useful tool for the construction and enumeration of meanders. Let P =(P +,P ) P n (M n ) be a meandric pair of noncrossing partitions of order n.. For any j N n we denote by [j] ± the unique block A P ± with j A, respectively. For any A P +, there exists a unique element r(a) :=r P (A) A for which there exists a (unique) sequence [r(a)] = C 0,C 1,...,C k = [1] + of distinct blocks C i P + C for which C i 1 C i for i = 1,...,k.Ifρ n (P) <n 1, then [1] + N n, and we can set A 1 := [1] +, and A 2 := [i 0 ] +, where i 0 := min ( ) N n \ A 1. Then, with j 0 := r P (A 2 ) and B 0 := [j 0 ], we define the reduced partitions P + := ( P + \{A 1,A 2 } ) {A 1 A 2 } and P := ( P \{B 0 } ) {B 0 \{j 0 }, {j 0 }}. With P denoting the pair of partitions (P +,P ), we can now show: Lemma 6. Suppose P is a pair of meandric noncrossing partitions of order n and rank ρ n (P) =k<n 1. Then P is a meandric pair of noncrossing partitions of order n and rank ρ n (P )=k +1and P P. Proof. (1) P + is noncrossing: Since A 1 and A 2 are neighbors we can choose i 1 A 1 such that i 2 := i 1 +1 A 2. Clearly, any ascending sequence a<b<c<dof crossing elements in the reduced partition P + induces an ascending sequence of crossing elements in P +. This is trivial if no elements of A 1 A 2 are involved. If the sequence a<b<c<d contains elements of A 1 A 2, then we can transform any crossing in A 1 A 2 into a crossing in A 1 or A 2, respectively, by inserting i 1 or i 2 in their proper place, as can be easily checked. (2) P is noncrossing: Clearly, any ascending sequence of crossing elements in P also crosses in P, since B 0 \{j 0 } B 0. (3) P is meandric: From the definition of P and by Theorem 1 follows immediately #P + +#P =(#P + 1) + (#P +1)=#P + +#P = n+1. Moreover, applying Theorem 1 again, we conclude that the associated graph Γ P of P is a tree. Clearly, the pair of noncrossing partitions (P +,P ) obtained from P by replacing B 0 by B 0 \{j 0 },{j 0 } in P decomposes

19 ON THE REPRESENTATION OF MEANDERS 19 into the two connected components P 1 and P 2 determined by the blocks A 1 and A 2, respectively. Therefore, considering that j 0 A 2, the reduced pair of noncrossing partitions P =(P +,P ), derived from (P +,P ) by replacing A 1,A 2 by their union A 1 A 2 in P +, is connected. Hence, again by Theorem 1, P N n. (4) Clearly, ρ n (P )=#P 1=(#P +1) 1=(#P 1) + 1 = ρ n (P)+1=k +1. (5) By construction, P + P +, and P P, which implies the assertion. The association P { P, if ρ n (P) <n 1, P, else defines a map R n : N n N n from the set of all meandric noncrossing partitions of order n into itself, which we will call the reduction map for N n. Example 3. Consider the pair of noncrossing partitions P = (1, 8, 12/ 2, 5, 6/3, 4/7/9, 11/10/13; 1, 13/2, 10, 11/3/4, 5, 7, 8/6/9/12) N 13. It changes under the reduction map R 13 into P =(1, 2, 5, 6, 8, 12/3, 4/7/ 9, 11/10/13; 1, 13/2, 10, 11/3/4, 7, 8/5/6/9/12) N 13. Note, that in this example i 0 =2, and j 0 =5. (Cf. Figure 6.) Clearly, R n is not surjective, since e.g. ˆ0 n / im(r n ). The restriction of R n to the set N n (k) :={P N n ρ n (P) =k} of all meandric noncrossing partitions of order n and rank k maps N n (k) onto a proper subset of N n (k +1)if k<n 2and n>2. This can be easily seen by considering for P N n the number c P := max{t N n N t [1] + }. In fact, if P im(r n ), then c P > 1. However, this condition is not sufficient, as the following example indicates: Example 4. P := (1, 2/3, 4; 1/2, 4/3) N 4 (2), butp / im(r 4 ), since {2} / P. Note that any pair of noncrossing partitions P N n (k) can be reduced to ˆ1 n by a suitable number of iterations of R n. In fact, we have as can be easily verified. R n 1 k n (P) =ˆ1 n,

20 20 REINHARD O. W. FRANZ FIGURE 6. The effect of the reduction map R n on the meandric tree G P associated with a pair of partitions P. (Cf. Example 3.) The original meandric tree G = G P The reduced meandric tree G = G P Example 5. Let M denote the meander given by P =(1/2, 5/3/4/6; 1, 2, 3/ 4, 5, 6), and M (i) the meander corresponding to P (i) = R6 i (P). Then, P (1) =(1, 2, 5/3/4/6; 1, 3/2/4, 5, 6) P (2) =(1, 2, 3, 5/4/6; 1/2/3/4, 5, 6) P (3) =(1, 2, 3, 4, 5/6; 1/2/3/4/5, 6) P (4) =(1, 2, 3, 4, 5, 6; 1/2/3/4/5/6).

21 ON THE REPRESENTATION OF MEANDERS 21 FIGURE 7. Reducing a meander M to 1 n by iterations of R n. M (4) M (3) M (2) M (1) M

22 22 REINHARD O. W. FRANZ The changes induced by R 6 in each step of the iteration to the meander M are depicted in Figure 7. We will now characterize the inverse images P N n (k) of a given element P =(P +,P ) N n (k +1)with respect to the reduction map for 0 k n 2 and c P > 1. To this end, choose an arbitrary non-empty subset A [1] P + of consecutive elements of [1] P + such that 1 / A, and A N cp. Then set P + := (P + \{[1] P + }) {A, [1]P + \ A}. Next, choose for any j 0 A for which {j 0 } P an arbitrary block {j 0 } B P pertaining to the connected component of [1] P + \ A with regard to (P +,P ) such that the partition P := (P \{{j 0 },B}) {B {j 0 }} is noncrossing. Let P := P (A, j 0,B):=(P +.P ). Then we can show: Theorem 5. Suppose P N n (k +1), 0 k<n 1, and suppose P is defined as above. Then, (1) P N n (k). (2) R n (P )=P. (3) For each inverse image P Rn 1 (P) of P there exists a suitable triple (A, j 0,B), selected as above, such that P = P (A, j 0,B). Proof. (1) By definition, P + and P are noncrossing. We use Theorem 1 to show that P is meandric. Clearly, #P ++#P =(#P + +1)+(#P 1) = #P + +#P = n +1. Note that, by definition, B P belongs to the connected component of [1] P + \ A while A lies in the other component of (P +,P ). Also note that j 0 A. Therefore, P =(P +,P ) obtained from (P +,P ) by replacing B,{j 0 } by B {j 0 } is connected. Finally, ρ n (P )= #P 1=(#P 1) 1=ρ n (P) 1=k. Hence, P N n (k). (2) Clearly, A 1 =[1] P + \ A, A 2 = A, r P (A 2 )=j 0, and B 0 = B {j 0 }. Therefore, by the definition of P and R n, it follows ( ((P+ (P + ) = \ [1] P + ) {A, A 1} ) ) \{A 1,A} {[1] P + } = P +, and ( ((P (P ) = \{{j 0 },B}) {B 0 } ) ) \{B 0 } {B,{j 0 }} = P, and thus R n (P )=P. (3) Now let P N n (k) such that R n (P )=P. We will show that we can select A, j 0,B suitably such that P = P (A, j 0,B). To this end, let A 1 := [1] P +, A 2 := [i 0 ] P +, i 0 := min(n n \ A 1 ), j 0 := r P (A 2 ), and

23 ON THE REPRESENTATION OF MEANDERS 23 B 0 := [j 0 ] P. First, note that by the definition of R n and the assumption R n (P )=P we have (10) and (11) (P + \{A 1,A 2 }) {A 1 A 2 } = P + (P \{B 0}) {B 0 \{j 0 }, {j 0 }} = P. Then, choose A := A 2. This implies [1] P + = A 1 A 2, and A 1 =[1] P + \ A. Then, with equation (10), we obtain P + =(P + \ [1] P + ) {A, [1]P + \ A} =(P+ \{A 1,A 2 }) {A 1,A 2 } = P+. Next, set j 0 := r P (A 2 ), B := B 0 \{j 0 }, and note that j 0 A, and B P because of (11). Therefore, utilizing equation (11) again, we obtain P =(P \{{j 0 },B}) {B {j 0 }} =(P \{B 0}) {B {j 0 }} = P. Clearly, the partition P is noncrossing. Moreover, B = B 0 \{j 0 } and A belong to different connected components of (P+ = P +,P ), since P is meandric, and j 0 A 2, B 0 P. We mention that the block B in the previous theorem can be selected without explicit check for connectivity and noncrossing of the partition P, which is crucial for the efficient construction of meanders using the reduction map: To this end, we first define for each P =(P +,P ) N n (k) a map γ P : P N n which assigns to every block B P the edge j N n which connects it with the root [1] P +, which can be done by recursion following the rank structure of N n using the reduction map R n. If P is the maximum of N n, i.e. P = ˆ1 =(N n, {{1}, {2},...,{n}}) N n (n 1), then we set γˆ1 ({j}) :=j for all j N n. Suppose, γ P has already been defined for all P N n (t) for t = n 1,n 2,...,k+1. We know that every element of N n (k) is the inverse image P (A, j 0,B)=(P +,P ) of some element P =(P +,P ) N n (k +1)for some suitable A [1] P +, j 0 A and B P with γ P (B) [1] P + \ A. Hence, we set for all C P { γ P (B) if C = B {j 0 } P γ P (C) or γ P (C) A := γ P (C) else. Then can be shown the following Corollary 1. With the notation from above, let P (A, j 0 ) denote the set of all B P pertaining to the connected component of [1] P + \ A with regard

24 24 REINHARD O. W. FRANZ to (P +,P ) such that the partition P := (P \{{j 0 },B}) {B {j 0 }} is noncrossing. Then P (A, j 0 )={B l P l =1, 2,...,n, γ P (B l ) [1] P + \ A}, where [ (σ 1 B l := P η )l ] P (j 0 ) and η := (1, 2,...,n) S n and σ P S n is the permutation associated with P (cf. Section 4) Constructing Meanders. The reduction map allows a very efficient constructing of the set N n without recurrence to meanders of lower order. Note that the sets ( N n (k), 0 k n 1, form a partition of N n. By construction, R n Nn (k) ) N n (k +1),ifk<n 1, and ( Nn (k +1) ), (12) N n (k) =Rn 1 which allows to construct the elements of N n of rank k provided those of rank k +1have already been constructed. Thus, since ˆ1 N n is the only meander of rank n 1, equation (12) allows the recursive construction of all the elements of N n, by successively creating the inverse images of N n (k) for k = n 1,...,1 using Theorem 5 and Corollary 1. In fact, considering the symmetry of the poset of meander, the construction can be stopped once k = (n 1)/2 has been reached, as can be easily verified. Clearly, only those elements P =(P +,P ) N n (k) possess inverse images for which c P := max{t N N t [1] + P } > 1 and for which there exists a singleton {j 0 } P such that 1 <j 0 [1] + P. For a more detailed description on this efficient and highly flexible construction method see [12] Counting Meanders. Recall that {N n (k) k =0,...,n 1} constitutes a partition of N n. Moreover, #N n (k) = #N n (n 1 k) for k =0,...,n 1, asn n is rank=symmetric. Hence #N n = 1 1 ( 1) n 2( ) n/2 1 #N n ( (n 1)/2 )+2 #N n (k), which allows to reduce the computation of #N n to the more easily accessible numbers #N n (k). We will briefly outline how those numbers can be determined. For more detail see [14]. We will first consider the special case k =1and count the number of meanders of order n and rank 1. Clearly, k=0 #N n (1) = #N n (n 2) = #R 1 n (ˆ1 n ).

25 ON THE REPRESENTATION OF MEANDERS 25 Therefore, we can use Theorem 5 and count the number of inverse images of ˆ1 n with respect to R n. To this end, we will consider for 1 j n 1 the sets N n (n 2,j):={P N n (n 2) #[1] P + = n j}, and set a nj := #N n (1,j), and a n := #N n (1). The meanders of order n and rank 1 can now be easily counted. For numerical results see Table 7.2. Theorem 6. Let n N and 1 j n 1. Then, (1) #N n (1,j)=(n j) 2 j. (2) #N n (1) = 1 12 n2 (n 2 1). Proof. (1) Let P := ˆ1 n = (1, 2,...n,;1/2/.../n) N n (n 1). We will use Theorem 5 and count the number of inverse images P of P with respect to R n. Since [1] P n = N n, for given 1 < j n 1, there are precisely n j distinct subsets 1 / A [1] P n of cardinality j consisting of consecutive elements of [1] P n. In order to count the possible choices for j 0 A and the corresponding admissible subsets B P, note that P = {{j 1 } j 1 N n }. Therefore, there are no constraints for choosing j 0 A, and B := {j 1 } pertains to the connected component of [1] P + \ A with regard to (P +,P ) if and only if j 1 [1] P + \A. Also note that replacing B = {j 1 }, {j 0 } by B {j 0 } = {j 0,j 1 } does not create any crossings in P. Hence, there are precisely j(n j) choices for j 0,B, which implies that #Rn 1 (ˆ1 n )=(n j) 2 j. (2) Clearly, we have n 1 n 1 #N n (1) = (n j) 2 j = j 2 (n j) j=1 j=1 n(n 1)(2n 1) = n 6 = n2 (n 2 1). 12 (n 1)2 n 2 4 In conclusion, we summerize some further results for counting meanders by rank, which we will present in more detail in [14]. We first note that it can be shown that =

26 26 REINHARD O. W. FRANZ TABLE 1. The first values for a nj and a n. n a n1 a n2 a n3 a n4 a n5 a n6 a n7 a n8 a n9 a n (13) l #N n (k) = 4k j=l j #N n 1 (k) for all n 3k +1and l =0,...,4k, which implies that the sequence of the (4k +1)-th differences of #N n (k) vanishes. Hence, #N n (k) satisfies a polynomial equation of degree 4k in n. In fact (14) #N n (k) = ( ) n + k +1 4k j=k+2 ( ) n j #N j (k), j as j #N j (k) =0for j =0,...,kand k+1 #N k+1 (k) =1. We have determined the cardinalities #N (j) j (k) (j = k +1,...4k) in equation (14) for k = 0,...,4 and derived the following counting formulae #N n (0) = 1, #N n (1) = n2 (n 2 1), #N n (2) = n2 (n 2 1)(n 2 4) p (n), 5 7 #N n (3) = n2 (n 2 1)(n 2 4)(n 2 9) p (n), 7 11 #N n (4) = n2 (n 2 1)(n 2 4)(n 2 9)(n 2 16) p (n), 11 13

27 ON THE REPRESENTATION OF MEANDERS 27 where the polynomials p k are given by p 0 (n) :=1 p 1 (n) :=1 p 2 (n) :=4n 2 7n 1 p 3 (n) := 227n n n n 940 p 4 (n) := 2 131n n n n n n We will call the polynomial p k the kth meandric polynomial. Note that the sequence ( #N n (1) ) of numbers of meanders of rank 1 defined by n #N n (1) = n2 (n 2 1) coincides with the well-known 4-dimensional pyramidal numbers (cf. [2],[19]). Figure 2 gives some computational results. 8. CONCLUSION The correspondence between meanders and pairs of permutations defined in Section 4 can be extended to a bijection between systems of meanders on surfaces and pairs of permutations [11]. REFERENCES [1] M. Aigner, Combinatorial Theory (Springer, Berlin, 1979). [2] O. D. Anderson, Find the next sequence, J. Rec. Math. 8 (No. 4, ), 241. [3] V. Arnol d, The Branched Covering of CP 2 S 4, Hyperbolicity and Projective Topology, Siberian Math. J. 29 (1988) 717. [4] C. Berge, Principles of Combinatorics (Academic Press, New York, London, 1971). [5] A. Björner, Shellable and Cohen-Macauly Partially Ordered Sets, Trans. Amer. Math. Soc. 260 No 1 (1980) [6] N. Dershowitz, Ordered Trees and Non-Crossing Partitions, Discrete Math. 62 (1986) [7] P.Di Francesco, O. Golinelli and E. Guitter, Meanders: A Direct Enumeration Approach, Nuclear Physics B [FS] 482 (1996) [8] P.Di Francesco, O. Golinelli and E. Guitter, Meanders: exact Asymptotics, Nuclear Physics B [FS] 570 (2000) [9] R. Franz, A Partial Order for the Set of Meanders, Ann. Comb. 2 (1998) [10] Franz, R., On the Structure of the Partially Ordered Set of Meanders, preprint. [11] R. Franz, Meanders on Surfaces, preprint. [12] R. Franz and B. Earnshaw, A Constructive Enumeration of Meanders, Ann. Comb. 6 (2002) [13] R. Franz, B. Earnshaw, J. Grout and J. Tripp, A Constructive Enumeration of Meanders: The Algorithm, preprint.

28 28 REINHARD O. W. FRANZ TABLE 2. Some computational results n #N n(0) #N n(1) #N n(2) #N n(3) #N n(4)

29 ON THE REPRESENTATION OF MEANDERS 29 [14] R. Franz and B. Earnshaw, Counting Meanders I, preprint. [15] M. Gardner, The combinatorial richness of folding a piece of paper, Scientific American 224, 6 (1971) [16] R. Guy, The Second Strong Law of Small Numbers, Math. Magazine 63, 1 (1990) [17] K. Hoffmann, K. Mehlhorn, P. Rosenstiehl and R. Tarjan, Sorting Jordan Sequences in Linear Time Using Level-linked Search Trees, Information and Control 68 (1986) 170. [18] I. Jensen, A transfer Matrix Approach to the Enumeration of Plane Meanders, J. Phys. A:Math. Gen. 33 (2000), [19] G. Kreweras, Traitemant simultane du Probleme de Young et du Probleme de Simon Newcomb, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), [20] J.E. Koehler, Folding a strip of stamps, J. Comb. 5 (1968) [21] K.H. Ko and A. Smolinksky, A Combinatorial Matrix in 3-Manifold Theory, Pacific. J. Math. 149 (1991) 319. [22] G. Kreweras, Sur les partitions noncroisées d un cycle, Discrete MatH. 1 (1972) [23] S.K. Lando and A.K. Zvonkin, Meanders, Selected translations, Selecta Math. Soviet. 11 (1992) [24] S.K. Lando and A.K. Zvonkin, Plane and Projective Meanders, Conference on Formal Power Series and Algebraic Combinatorics, Bordeaux, 1991, in: Theoret. Comput. Sci. 117(1 2) (1993) [25] E. Lucas, Theorie des Nombres, Vol 1 (Gauthier-Villars, Paris 1891) 120. [26] W.F. Lunnon, A Map Folding Problem, Math. of Comp. 22 (1968) [27] W.F. Lunnon, Multidimensional Map Folding, The Comp. J. 14, No. 1 (1971) [28] Sainte-Lague, Memorial des Sciences Mathematiques, Gauthier-Villars Fasc 18 (1926) [29] Sainte-Lague, Avec des Nombres et des Lignes (Vuibert Paris, 1937), [30] R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math. 98 (1991) [31] N.J.A. Sloane, A Handbook of Integer Sequences (Academic Press, 1973). [32] R.P. Stanley, Enumerative Combinatorics Vol. 1 Wadsworth, Belmont, CA, 1986). [33] R.P. Stanley, Parking functions and noncrossing partitions, The Wilf Festschrift, Electron.J.Combin. 4 no. 2 (1997). [34] J. Touchard, Contributions a l etude du probleme des timbres poste, Can J. of Math. 2 (1950) REINHARD O. W. FRANZ, COLLEGE OF ENGINEERING AND TECHNOLOGY,BRIGHAM YOUNG UNIVERSITY, PROVO, UT U.S.A. address: franz@math.byu.edu