Meanders and Motzkin Words

A. Panayotopoulos and P. Tsikouras Deptartment of Informatics University of Pireaus Karaoli & Dimitriou 80 18534 Pireaus Greece antonios@unipi.gr pgtsik@unipi.gr

Abstract

We study the construction of closed meanders and systems of closed meanders, using Motzkin words with four letters. These words are generated by applying binary operation on the set of Dyck words. The procedure is based on the various kinds of intersection of the meandric curve with the horizontal line.

1 Introduction

Among various efforts to study and to generate meanders, Jensen [4] has used sequences related to the intervals between the crossing points along the horizontal line, Franz and Earnshaw [3] have used noncrossing partitions, whereas the authors [8] as well as Barraud et al. [1] have used planar permutations which follow the meandric curve.

This paper refers to the study and construction of closed meanders and systems of closed meanders, using Motzkin words.

The following definitions and notation refer to notions that are necessary for the development of the paper.

A word u ∈ {a, ā}^{∗} is called a Dyck word if |u|_{a} = |u|ā and for every factorization u = pq we have |p|_{a} ≥ |p|ā where |u|_{a}, |p|_{a} (resp. |u|ā, |p|ā) denote the number of occurrences of a (resp. ā) in the words u,p.

A word w ∈ {a, ā, x, y}^{∗} is called a Motzkin word if |w|_{a} = |w|ā and for every factorization w = pq we have |p|_{a} ≥ |p|ā, or equivalently if the word obtained by deleting every occurence of x, y from w is a Dyck word of {a, ā}^{∗}.

Let D_{2n} denote the set of all Dyck words of length 2n. It is well known that the cardinality of D_{2n} equals to the Catalan number C_{n} = \frac{1}{n+1} \binom{2n}{n} (A000108); Panayotopoulos and Sapounakis [6] have presented a construction of D_{2n}.

Let u = u_{1}u_{2} ⋯ u_{2n} with u ∈ D_{2n}. Two indices i, j such that i < j, u_{i} = a, u_{j} = ā are called conjugates with respect to u if j is the smallest element of {i+1, i+2, …, 2n} for which the subword u_{i}u_{i+1} ⋯ u_{j} is a Dyck word.

There exists a bijection between D_{2n} and N_{2n}, since for each u ∈ D_{2n} we can determine its corresponding nested set S_{u} ∈ N_{2n} as follows : {i, j} ∈ S_{u} if and only if i, j are conjugate indices with respect to u.

For example, the nested set {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}} corresponds to the Dyck word u = a a a ā ā ā a ā a ā.

We also recall that if we denote by dom S all the elements of N^{∗} that belong to some pair of a nested set of pairs S, we say that two nested set S_{1}, S_{2} are matching if dom S_{1} = dom S_{2} and dom A = dom B, A ⊆ S_{1}, B ⊆ S_{2} imply that either A = B = ∅ or dom A = dom S_{1}.

Furthermore, we call B ⊆ dom S complete if for every a ∈ B with {a, b} ∈ S, we have b ∈ B. We write S/B = {{a, b} ∈ S : a ∈ B}. For every two nested sets S_{1}, S_{2} with dom S_{1} = dom S_{2} that are not matching, there exists a partition B_{1}, B_{2}, …, B_{k} of dom S_{1} with B_{i} complete, such that the sets S_{1}/B_{i}, S_{2}/B_{i}, i ∈ [k] are matching; we then call S_{1}, S_{2} k-matching [7].

Geometrically, if we draw two matching nested sets, one above and the other underneath the horizontal axis, they form a simple, closed curve, whereas two kmatching nested sets create k such curves; (see Figures 1 and 2).

In section 2 we define the m-Motzkin words. To each such word corresponds a pair of nested sets which are either matching or k-matching. The set of m-Motzkin words is partitioned into classes of either two or four elements.

In section 3 we prove that there exists a bijection between the set of closed meanders and the set of m-Motzkin words which correspond to matching nested sets. Using this bijection, we can generate closed meanders from m-Motzkin words.

In section 4 we extend the above results to systems of meanders and we present a recursive generation of these systems.

2 m-Motzkin words

For every pair u = u_{1}u_{2} ⋯ u_{2n}, u′ = u′_{1}u′_{2} ⋯ u′_{2n} of elements of D_{2n}, we define u∘u′ to be the word w = w_{1}w_{2} ⋯ w_{2n}, with

w_{i} = \{ \begin{matrix}a, & if u_{i} = u′_{i} = a; \\ ā, & if u_{i} = u′_{i} = ā; \\ b, & if u_{i} = a, u′_{i} = ā; \\ b̄, & if u_{i} = ā, u′_{i} = a. \end{matrix}

For example, from u = a a a ā ā ā a ā a ā and u′ = a ā a a ā a ā a ā ā we obtain u∘u′ = a b a b̄ ā b̄ b b̄ b ā.

We write Ŵ_{2n} = {w : w = u∘u′, u, u′ ∈ D_{2n}}.

Proposition 2.1 If w ∈ Ŵ_{2n} then w is a Motzkin word of {a, ā, b, b̄}^{∗}, with |w|_{b} = |w|b̄.

Proof : Let I_{1} = {i ∈ [2n] : u_{i} = u′_{i} = a}, I_{2} = {i ∈ [2n] : u_{i} = a, u′_{i} = ā}, I_{3} = {i ∈ [2n] : u_{i} = ā, u′_{i} = a}, I_{4} = {i ∈ [2n] : u_{i} = u′_{i} = ā}. Given that u, u′ ∈ D_{2n} we have that: |I_{1}| + |I_{2}| = |I_{3}| + |I_{4}| and |I_{1}| + |I_{3}| = |I_{2}| + |I_{4}|; so we get |I_{3}| = |I_{2}| and |I_{1}| = |I_{4}|, i.e. |w|_{b} = |w|b̄ and |w|_{a} = |w|ā.

Let now z be the word that we obtain by deleting every occurrence of b, b̄ in w. Obviously |z|_{a} = |w|_{a} = |w|ā = |z|ā. In order to show that z is a Dyck word,

we must also have |s|_{a} ≥ |s|ā, for every factorization z = st. This is true, since if |s|_{a} < |s|ā, for some such factorization, then for at least one of the words u, u′ we would have a factorization pq with |p|_{a} < |p|ā, contradicting the assumption that both u and u′ are Dyck words. □

We call the elements of Ŵ_{2n} meandric Motzkin words (or m-Motzkin words) of length 2n.

Let now w = w_{1}w_{2} ⋯ w_{2n}, with w ∈ Ŵ_{2n}. From w we obtain two words r = r_{1}r_{2} ⋯ r_{2n}, r′ = r′_{1}r′_{2} ⋯ r′_{2n} of {a, ā}^{∗}, with

r_{i} = \{ \begin{matrix}a, & if w_{i} = a or b; \\ ā, & if w_{i} = ā or b̄, \end{matrix} r′_{i} = \{ \begin{matrix}a, & if w_{i} = a or b̄; \\ ā, & if w_{i} = ā or b. \end{matrix}

We call r and r′ relatives of w.

Practically, in order to obtain r we change each occurrence of b, b̄ of w into a, ā respectively, whereas in order to obtain r′ we change b, b̄ into ā, a respectively.

Proposition 2.2 Let w ∈ Ŵ_{2n} with w = u∘u′, u, u′ ∈ D_{2n} and let r, r′ be its relatives. Then r = u and r′ = u′.

Proof : Let w_{i} = a (resp. ā). Then r_{i} = a = u_{i} and r′_{i} = a = u′_{i} (resp. r_{i} = ā = u_{i} and r′_{i} = ā = u′_{i}). Let now w_{i} = b (resp. b̄). Then r_{i} = a = u_{i} and r′_{i} = ā = u′_{i} (resp. r_{i} = ā = u_{i} and r′_{i} = a = u′_{i}).

So, we realize that in every case the elements of r and u as well as the elements of r′ and u′ coincide, giving the required result. □

From the bijection between the sets D_{2n} × D_{2n} and Ŵ_{2n} that we have established, we obviously get the following relation :

|Ŵ_{2n}| = (C_{n})^{2} .

Notice that from the word u∘u′ we immediately obtain the word u′∘u, by interchanging the letters b and b̄. So, in order to generate the set Ŵ_{2n} it is actually enough to construct half of its elements.

So, by the above procedure we also create for each w ∈ Ŵ_{2n} two nested sets S_{w}, S′_{w} on [2n] corresponding to the words r, r′ ∈ D_{2n}.

We denote with W_{2n} (resp W_{2n}^{k}) the set of al the words w ∈ Ŵ_{2n} for which S_{w}, S′_{w} are matching (resp. k-matching).

For example, the word w = a b a b̄ ā b̄ b b̄ b ā is an m-Motzkin word, for which we have r = a a a ā ā ā a ā a ā and r′ = a ā a a ā a ā a ā ā.

The corresponding nested sets S_{w} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}} and S′_{w} = {{1, 2},{3, 10}, {4, 5}, {6, 7}, {8, 9}} are matching.

Similarly, the word w = a a b b̄ ā b̄ a ā b ā is a m-Motzkin word, for which we have r = a a a ā ā ā a ā a ā and r′ = a a ā a ā a a ā ā ā. The corresponding nested sets

S_{w} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}}

and

S′_{w} = {{1, 10}, {2, 3}, {4, 5}, {6, 9}, {7, 8}}

are 3-matching, with B_{1} = {1, 6, 9, 10}, B_{2} = {2, 3, 4, 5} and B_{3} = {7, 8}, thus determining the matching nested sets : S_{w}/B_{1} = {{1, 6}, {9, 10}}, S_{w}/B_{2} = {{2, 5}, {3, 4}}, S_{w}/B_{3} = {{7, 8}} S′_{w}/B_{1} = {{1, 10}, {6, 9}}, S′_{w}/B_{2} = {{2, 3}, {4, 5}}, S′_{w}/B_{3} = {{7, 8}}.

It is easy to obtain the following result.

Proposition 2.3 If w ∈ W_{2n}^{k} then there exist k subwords w^{j} ∈ W_{2s_{j}}, j = 1, 2, …, k with s_{1} + s_{2} + ⋯ + s_{k} = n which can be recognized in w.

For example, in the word w = a a b b̄ ā b̄ a ā b ā ∈ W_{10}^{3}, we recognize the subwords w^{1} = w_{1}w_{6}w_{9}w_{10} = a b̄ b ā ∈ W_{4}, w^{2} = w_{2}w_{3}w_{4}w_{5} = a b b̄ ā ∈ W_{4} and w^{3} = w_{7}w_{8} = a ā ∈ W_{2}.

We continue by introducing three internal operations in the set Ŵ_{2n}:

For w ∈ Ŵ_{2n}, we define the words w^{}, w^{−} and w^{+} as follows:

w_{i}^{}= w̄_{2n+1−i} (where ¯ w̄_{j} = w_{j})

w_{i}^{−} = \{ \begin{matrix}w_{i}, & if w_{i} ∈ \{a, ā\}; \\ b, & if w_{i} = b̄; \\ b̄, & if w_{i} = b, \end{matrix} w_{i}^{+} = \{ \begin{matrix}ā, & if w_{2n+1−i} = a; \\ a, & if w_{2n+1−i} = ā; \\ w_{2n+1−i}, & if w_{2n+1−i} ∈ \{b, b̄\}, \end{matrix}

for every i ∈ [2n]. We may call these operations mirror, overturn and mirror-overturn respectively.

For example, if w = a b a b̄ ā b̄ b b̄ b ā then w^{}= a b̄ b b̄ b a b ā b̄ ā, w^{−} = a b̄ a b ā b b̄ b b̄ ā and w^{+} = a b b̄ b b̄ a b̄ ā b ā.

It is obvious that for any w ∈ Ŵ_{2n}, the words w^{}, w^{−} and w^{+} also belong to Ŵ_{2n}. We have that w^{}= w (resp. w^{+} = w) iff w^{+} = w^{−} (resp. w^{−} = w^{}) as well as that w^{−} ≠ w and w^{+} ≠ w^{}. We thus obtain the following result.

Proposition 2.4 The set Ŵ_{2n} can be partitioned into classes of either two or four elements.

Let A_{2n} = {w ∈ Ŵ_{2n} : w_{2} = a, w_{2n−1} = ā} and B_{2n} = {w ∈ Ŵ_{2n} : w_{2} = b̄}. By the previous properties of w^{}, w^{−} and w^{+} we have the following proposition.

Proposition 2.5 i) If w ∈ A_{2n}, then w^{}, w^{−}, w^{+} ∈ A_{2n}. ii) If w ∉ A_{2n}, then at least one of the words w, w^{}, w^{−}, w^{+} belongs to B_{2n}.

From the previous results it is clear that in order to construct Ŵ_{2n} it is enough to have A_{2n} and B_{2n}. In order now to generate each element w = u�u′ of A_{2n} (resp. B_{2n}), it is enough to consider only the words u = u_{1}u_{2} ⋯ u_{2n}, u′ = u′_{1}u′_{2} ⋯ u′_{2n} of D_{2n} with u_{2} = u′_{2} = a and u_{2n−1} = u′_{2n−1} = ā (resp. u_{2} = ā, u′_{2} = a).

3 Meanders

We recall that a closed meander of order n is a closed self avoiding curve, crossing an infinite horizontal line 2n times (A005315).

Let M_{2n} be the set of all closed meanders of order n.

As opposed to previous papers [1], [8], the study of meanders will follow here the order of the crossings of the horizontal line rather than the meandric curve itself.

It is clear that if μ ∈ M_{2n}, the lines above (resp. underneath) the horizontal line uniquely define a nested set U_{μ} (resp. L_{μ}) on [2n] with U_{μ}, L_{μ} being matching and

Figure 1: A closed meander of order 5

conversely two matching nested sets U_{μ} and L_{μ} uniquely define the meander μ. This allows us to actually identify a meander μ ∈ M_{2n} to a pair (U_{μ},L_{μ}) of nested sets of [2n].

For example, for the closed meander μ of Figure 1 we have: U_{μ} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}}, L_{μ} = {{1, 2}, {3, 10}, {4, 5}, {6, 7}, {8, 9}}.

To each meander of M_{2n} corresponds a unique word of W_{2n}. Intuitively, this correspondence becomes obvious when we assign the letters a, ā, b, b̄ to the various kinds of intersection opening, closing, proceeding upwards, proceeding downwards respectively, occurring along the horizontal line.

So, the word w = a b a b̄ ā b̄ b b̄ b ā corresponds to the closed meander of Figure 1.

In order to develop formally these ideas we need the following result, obtained by considering all the possible orderings for the elements i, j, h of the pairs {i, j} ∈ U_{μ} and {i, h} ∈ L_{μ}.

To every μ ∈ M_{2n} corresponds a unique word w ∈ W_{2n}, with

w_{i} = \{ \begin{matrix}a, & if i < j, h; \\ ā, & if h, j < i; \\ b, & if h < i < j; \\ b̄, & if j < i < h, \end{matrix}

where {i, j} ∈ U_{μ} , {i, h} ∈ L_{μ}.

So, from the nested sets U_{μ}, L_{μ} of the previous example we create again the word w = a b a b̄ ā b̄ b b̄ b ā.

Conversely, to every word w ∈ W_{2n} with S_{w}, S′_{w} matching, corresponds a unique meander μ ∈ M_{2n} with U_{μ} = S_{w}, L_{μ} = S′_{w}.

From the above, we have the following result.

Proposition 3.1 There exists a bijection between the sets M_{2n} and W_{2n}.

In order to determine U_{μ} and L_{μ} (and hence construct the meander μ ∈ M_{2n}) we use the notion of conjugate indices of a Dyck word. So, given a word w ∈ W_{2n}, we create its relatives r, r′ and we find the conjugate indices of these Dyck words, which indicate the pairs of U_{μ} and L_{μ} respectively.

We recall that a pair {a, b} of a nested set S is called short pair if there is no c ∈ dom S with either a < c < b or b < c < a, [7]. We have the following result.

Proposition 3.2 Each digram aā, ab̄, bā, bb̄ (resp. aā, ab, b̄ā, b̄b) of a word w ∈ W_{2n} corresponds to a short pair of U_{μ} (resp. L_{μ}) in the associated meander μ.

So, we can also determine the meander μ ∈ M_{2n} by repetitively contracting the given word w ∈ W_{2n}, using each time propositions 3.1 and 3.2.

So for w = a b a b̄ ā b̄ b b̄ b ā, we have:

\begin{aligned}w = & a & b & a & b̄ & ā & b̄ & b & b̄ & b & ā \\ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \end{aligned}

\begin{matrix}a & b & ā & b̄ \\ 1 & 2 & 5 & 6 \end{matrix}

\begin{matrix}a & b̄ \\ 1 & 6 \end{matrix}

giving U_{μ} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}}.

Similarly, we have

\begin{aligned}w = & a & b & a & b̄ & ā & b̄ & b & b̄ & b & ā \\ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \end{aligned} ; \begin{matrix}a & ā \\ 3 & 10 \end{matrix}

giving L_{μ} = {{1, 2}, {3, 10}, {4, 5}, {6, 7}, {8, 9}}.

We thus finally get the meander μ of Figure 1 again.

Let μ ∈ M_{2n} and w ∈ W_{2n} its corresponding word. If we draw the meanders μ^{}, μ^{−}, μ^{+} that correspond to the words w^{}, w^{−}, w^{+} we realize that the above operations define meanders symmetric to the meander μ that corresponds to w, with respect to a vertical axis, to the horizontal line and to their intersection respectively.

It is easy to check that: {i, j} ∈ U_{μ^{}}iff {2n + 1 − i, 2n + 1 − j} ∈ U_{μ}, {i, j} ∈ L_{μ^{}}, iff {2n + 1 − i, 2n + 1 − j} ∈ L_{μ}, U_{μ^{−}} = L_{μ}, L_{μ^{−}} = U_{μ}, U_{μ^{+}} = L_{μ^{}}, L_{μ^{+}} = U_{μ^{}}.

Hence, according to Proposition 2.4 we have the following result.

Proposition 3.3 The set M_{2n} can be partitioned into classes of either two or four elements.

4 Systems of meanders

We can extend the definition of closed meanders to systems of closed meanders with k components (or k-meanders) by allowing configurations with k disconnected meanders [5]. We will denote the set of all k-meanders of order n with M_{2n}^{k}, k ∈ {2, 3, …, n}.

Figure 2: A 3-meander of order 5

Obviously, like in the case of meanders, a k-meander ν also determines the corresponding nested sets U_{ν}, L_{ν} that are now k-matching.

For example, for the 3-meander ν of Figure 2 we have: U_{ν} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}, {9, 10}} L_{ν} = {{1, 10}, {2, 3}, {4, 5}, {6, 9}, {7, 8}}.

We can still assign the letters a, ā, b, b̄ to the various kinds of intersection, thus creating the corresponding word of W_{2n}^{k}.

So, the word w = a a b b̄ ā b̄ a ā b ā corresponds to the 3-meander of Figure 2. It is easy to check that if we refer to meanders of M_{2n}^{k} instead of M_{2n}, to W_{2n}^{k}

instead of W_{2n} and to k-matching instead of matching nested sets, we can apply propositions 3.1, 3.2 and 3.3 to k-meanders.

So, similarly to proposition 3.1, there exists a bijection between the sets M_{2n}^{k} and W_{2n}^{k}, i.e., to every ν ∈ M_{2n}^{k} corresponds a unique word w ∈ W_{2n}^{k} obtained by the formula for w_{i}.

Conversely, to every w ∈ W_{2n}^{k} with S_{w}, S′_{w} k-matching, corresponds a unique system of meanders ν ∈ M_{2n}^{k} with U_{ν} = S_{w}, L_{ν} = S′_{w}.

P. Di Francesco et al. [2] have given formulae for the cardinality of M_{2n}^{k}, for k = n−3, n−2, n−1, whereas for k = n we have |M_{2n}^{n}| = C_{2n}, given that W_{2n}^{n} = D_{2n}.

Similarly to proposition 3.2, we can now determine the system of meanders ν ∈ M_{2n}^{k} from the word w ∈ W_{2n}^{k}.

Let now S be a member of the set N_{2n} of the nested sets of pairs on [2n]; let {a, d}, {b, c} ∈ S with a < b < c < d and such that for every {e, f} ∈ S with e < b < f we have e ≤ a; then {a, d} (resp. {b, c}) is called father (resp. child) of {b, c} (resp. {a, d}). We call two elements {i, j}, {k, l} of S brothers if the have the same father, or if they have no father.

We define two operations in N_{2n} as follows:

If {b, c} and its father {a, d} belong to S ∈ N_{2n} with a < b < c < d, then σ(S; a, b) is the set obtained if we replace the pairs {a, d} and {b, c} with the pairs {a, b} and {c, d}. It is obvious that σ(S; a, b) ∈ N_{2n} and that {a, b} and {c, d} are brothers in σ(S; a, b).

If {a, b}, {c, d} are brothers in S ∈ N_{2n}, with a < b < c < d, then τ(S; a, c) is the set obtained if we replace the pairs {a, b} and {c, d} with the pair {a, d} and {b, c}. It is obvious that τ(S; a, c) ∈ N_{2n} and that {a, d} is the father of {b, c} in τ(S; a, c).

The above definitions imply that if {a, d} is the father of {b, c} in the set S ∈ N_{2n}, then τ(σ(S; a, b); a, c) = S, whereas if {a, b}, {c, d} are brothers, then

σ(τ(S; a, c); a, b) = S.

We also have the following result.

Proposition 4.1 Let ν ∈ M_{2n}^{k}. If the father {a, d} and the child {b, c} (resp. the brothers {a, b}, {c, d}) of U_{ν} belong to the same component of ν then the set U = σ(U_{ν}; a, b) (resp. U = τ(U_{ν}; a, c)) and L_{ν} are (k + 1)-matching, thus defining a meander ξ ∈ M_{2n}^{k+1}, whereas if they belong to different components of ν, then ξ belongs to M_{2n}^{k−1}.

It is obvious that the above result still holds if we interchange L with U.

Proposition 4.1 is important since it enables us to recursively construct the sets M_{2n}^{k}, k = j + 1, j + 2, …, n if the set M_{2n}^{j} is known for some j ∈ [n − 1].

For example, if μ ∈ M_{10} is the meander of Figure 1, we have that the sets U_{ν} = U_{μ} and L_{ν} = τ(L_{μ}; 6, 8) = {{1, 2}, {3, 10}, {4, 5}, {6, 9}, {7, 8}} determine a meander ν ∈ M_{10}^{2}; a second application of proposition 4.1 gives U_{ξ} = U_{ν} and

L_{ξ} = τ(L_{ν}; 1, 3) = {{1, 10}, {2, 3}, {4, 5}, {6, 9}, {7, 8}}

which determine the meander ξ ∈ M_{10}^{3} of Figure 2.

References

[1] J. Barraud, A. Panayotopoulos and P. Tsikouras, Properties of closed meanders, Ars Combin. 67 (2003), 189–197.

[2] P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, Math. Comput. Modelling 26 (1997), 97–147.

[3] R. O. W. Franz and B. A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), 7–17.

[4] I. Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33 (2000), 5953–5963.

[5] S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoret. Comput. Sci. 117 (1993), 227–241.

[6] A. Panayotopoulos and A. Sapounakis, On binary trees and Dyck paths, Math. Inf. Sci. Hum. 131 (1995), 39–51.

[7] A. Panayotopoulos and P. Tsikouras, The multimatching property of nested sets, Math. Sci. Hum. 149 (2000), 23–30.

[8] A. Panayotopoulos and P. Tsikouras, Properties of meanders, J. Combin. Math. Combin. Comput., 46 (2003), 181–190.

2000 Mathematics Subject Classification: Primary 05A15; Secondary 05A18. Keywords: Meanders, Dyck words, Catalan numbers, Motzkin words.

(Concerned with sequences \underline{A000108} and \underline{A005315}.)

Received April 21 2003; revised version received January 20 2004. Published in Journal of Integer Sequences, February 10 2004.

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