The Number of Convex Permutominoes

Transcription

1 The Nuber of Convex Perutoinoes Paolo Boldi, Violetta Lonati, Roberto Radicioni, and Massio Santini Dipartiento di Scienze dell Inforazione Università degli Studi di Milano Abstract. Perutoinoes are polyoinoes defined by suitable pairs of perutations. In this paper we provide a forula to count the nuber of convex perutoinoes of given perieter. To this ai we define the transfor of a generic pair of perutations, we characterize the transfor of any pair defining a convex perutoino, and we solve the counting proble in the transfored space. 1 Introduction A polyoino (also known as lattice anial is a finite collection of square cells of equal size arranged with coincident sides. In this paper we consider a special class of polyoinoes, naely the perutoinoes, that we define in a purely geoetric way. Actually, the ter perutoino arises fro the fact that this object can be defined by a diagra on the plane representing a pair of perutations. Such diagras were introduced in [8] as a tool to study Schubert varieties and used in [6] (where the ter perutaoino appeared for the first tie and [7] in relation to Kazhdan-Lusztig R-polynoials. Counting the nuber of polyoinoes and perutoinoes is an interesting cobinatorial proble, still open in its ore general for; yet, for soe subclasses of polyoinoes, exact forulae are known. For instance, the nuber of convex polyoinoes (i.e., whose intersection with any vertical or horizontal line is connected of given perieter has been obtained in [2], whereas the enueration proble for soe subclasses of convex perutoinoes has been solved in [5]. In this paper, we provide an explicit forula for the nuber of convex perutoinoes of a given perieter. Our counting technique is based on two basic facts. First, the boundary of every convex perutoino can be decoposed into four subpaths describing, in this order, a down/rightward, up/rightward, up/leftward, down/leftward stepwise oveent. Second, for each abscissa (ordinate there is exactly one vertical (horizontal segent in the boundary with that coordinate. Actually, these two constraints hold not only for the boundary of convex perutoinoes, but for a larger class of circuits we call adissible: in Section 3 we describe adissible circuits and we obtain their nuber A n in Section 5. In Section 4 we characterize adissible circuits that do not define a perutoino: again we obtain their nuber B n in Section 5. As a consequence, we get the nuber of convex perutoinoes as the difference A n B n. 2 Preliinaries In this section, we shall recall soe basic definitions and properties of polyoinoes, perutoinoes and generating functions. 2.1 Polyoinoes and perutoinoes A cell is a closed subset of R 2 of the for [a,b] [a + 1,b + 1], where a,b Z; we shall identify such a cell with the pair (a,b. Let us define a binary relation of adjacency between cells by letting (a,b (a,b if and only if a = a and b b = 1, or a a = 1 and b = b. A subset P of R 2 is a This work is partially supported by the MIUR PRIN Project Autoi e linguaggi forali: aspetti ateatici e applicativi.

2 polyoino if and only if it is a finite nonepty union of cells that is connected by adjacency, i.e., such that if (a,b,(a,b P then there exist (a 1,b 1,...,(a k,b k P such that (a,b = (a 1,b 1 (a 2,b 2 (a k,b k = (a,b. See Figure 1 (a for an exaple. A polyoino is defined up to translations; without loss of generality, we assue that the lowest leftost vertex of the ininal bounding rectangle of the polyoino is placed at the point (1,1. Special types of polyoinoes P are the following: P is row-convex if and only if (a,b,(a,b P and a a a iply (a,b P; P is colun-convex if and only if (a,b,(a,b P and b b b iply (a,b P; P is convex if and only if it is both row- and colun-convex; P is directed if and only if it contains at least one of the corner cells of its inial bounding rectangle; P is parallelogra if and only if it is convex and contains at least a pair of opposite corner cells of its inial bounding rectangle (e.g., both the lower-left and upper-right cells. The (topological border of a polyoino P is a disjoint union of siple closed curves; in particular, if there is only one curve, we say that P has no holes: all polyoinoes in this work will have no holes. The border is a siple closed curve ade of alternating vertical and horizontal nontrivial segents whose endpoints (vertices have integral coordinates; conversely, every such a closed curve is the border of a polyoino without holes, so we shall freely identify polyoinoes with their borders. We say that P is a perutoino of size n if and only if its inial bounding rectangle is a square of size n 1, and the border of P has exactly one vertical segent of abscissa z and one horizontal segent of ordinate z, for every z {1,...,n}. Notice that a convex perutoino of size n has perieter 4(n 1. In order to handle polyoinoes we introduce the following definitions. A (stepwise siple path is a sequence P 1 = (x 1,y 1, P 1 = (x 1,y 1, P 2 = (x 2,y 2, P 2 = (x 2,y 2..., P = (x,y, P = (x,y of distinct points with integer coordinates such that, for all i {1,...,}, x i = x i, and y i = y i+1 if i < ; notice that the segents P i P i are vertical, whereas the segents P i P i+1 are horizontal. More generaly, a path is a sequence of points P 1, P 1,..., P k, P k such that, for soe k, P 1, P 1,..., P, P is a siple path, and for all i >, P i = P i and P i = P i. A circuit is a siple path such that y = y 1; when dealing with circuits, we shall iplicitly assue that the subscripts are treated odulo ; so, for exaple P +1 is just P 1. A point is a (self-crossing point of a siple path if and only if it is the intersection of two segents, say P i P i and P j P j+1; we also say that the path has a crossing at indices (i, j. Clearly, visiting the border of a polyoino P counter-clockwise and starting fro the highest vertex of the leftost edge, we identify a circuit without crossing points: we call it the boundary of P and denote it by P 1 = A,P 1,P 2,P 2,...,P,P (see Figure 1 (a. Notice that if P is a perutoino, then = n. In particular we consider four special points in the boundary of any polyoino P: let A = P 1 be the highest vertex of the leftost edge, B be the leftost vertex of the lowest edge, C be the lowest vertex of the rightost edge, D be the rightost of the highest edge (see Figure 1 (b. Notice that, if P is convex, then the subsequence of vertices between A and B (B and C, C and D, D and P, respectively is a path directed down/rightward (up/rightward, up/leftward, down/leftward, respectively; see Figure 1 (c. 2.2 Generating functions The generating function f(z of the sequence {a n } n is defined as [9] f(z = n a n z n ; it is well-known that z f (z = n ( n na n z n and f(z g(z = n b n k z n k=0a n,

3 P 1 P 11 P 10 P 9 P 8 A D D P 1 P 2 P 9 P 3 P 11 P 10 P 7 P 8 C P 2 P 6 P 7 A P 3 P 4 P 5 C P 4 P 5 P 6 B (a (b (c Fig. 1. (a The boundary of a polyoino. (b The extree points of a polyoino. (c The extree points of a convex polyoino. B where g(z = n b n z n. Soe exaples of generating functions that we will need in the following are: ( 1 2n = 1 4z z n n n 1 1 4z = 4 n z n. n As a consequence of the previous facts, we have that ( 2k k n k=0 ( 2(n k n k 3 Perutoinoes, perutations, and transfor = 4 n. (1 In this section we illustrate the relationship between the set of perutoinoes of size n and the set Π n = {(σ,τ σ,τ S(n,σ(x τ(x for every x, and σ(1 > τ(1}. Consider a perutoino P of size n and let P 1,P 1,P 2,P 2,...,P n,p n be its boundary. By definition, for any z {1,...,n} there is exactly one index i such that P i and P i have abscissa z and there is exactly one index j such that P j and P j+1 have ordinate z. Thus, a perutoino P of size n uniquely deterines a pair of perutations (σ,τ Π n (which we call the perutation pair of P: σ(x and τ(x are defined as the respective ordinates of the (unique points P i and P i with abscissa x. In particular, observe that A = P 1 = (1,σ(1, B = (τ 1 (1,1, C = (n,σ(n, D = (τ 1 (n,n. Conversely, a pair (σ,τ Π n does not always define a perutoino. However, one can always consider the set of points (i {1,...,n} where, for every i < n, T i = (x i,τ(x i and S i = (x i,σ(x i, x 1 = 1, x i = σ 1 (τ(x i 1, We define the path of (σ,τ as the path S 1,T 1,S 2,T 2,...,S n,t n. Notice that this path needs not be siple, as Figure 2 (c illustrates. However, if the 2n points are all distinct, the path is indeed a circuit with exactly one vertical (and horizontal segent for every abscissa (and ordinate, see Figure 2 (a; yet, the circuit ay contain crossing points (see Figure 2 (b.

4 (a (b (c Fig. 2. (a The perutoino of size n = 7 defined by σ = (5,7,4,1,6,3,2 and τ = (4,5,1,2,7,6,3 (squares represent σ, whereas lozenges represent τ. (b The path of σ = (3,2,1,4,7,5,6 and τ = (2,1,7,3,6,4,5, which does not define a perutoino. (c The path of σ = (3,2,1,4,6,5,7 and τ = (2,1,3,7,5,4,6, which does not define a perutoino of size n = 7. Reark 1. A pair of perutations in Π n is the perutation pair of a perutoino P if and only if its path has exactly 2n distinct vertices and has no crossing points. In this case its path coincides with the boundary of P, that is S i = P i and T i = P i for every i. We notice that this reark is actually the definition of perutoino as introduced in [6]. Indeed, our definition of path of a pair of perutation recalls the geoetric contruction used in [6], even though there are soe differences (for instance in the case of Figure 2 (c. We now introduce a ap F n : Π n Φ n where Φ n is the set of pairs of endofunctions of {1,...,n}. For any pair (σ,τ Π n we set F n (σ,τ = (v,h where v(1 = 1, v(i+1 = σ 1 (τ(v(i h(1 = τ(1, h(i+1 = τ(σ 1 (h(i for every i {1,...,n 1}. The pair (v,h is called the transfor of (σ,τ; Figure 3 shows an exaple of perutoino and the transfor (v,h of its perutation path (σ,τ. The transfor F n has the following geoetric interpretation: v(i is the abscissa of the i-th vertical edge along the path of (σ, τ, whereas h(i is the ordinate of the i-th horizontal edge along the sae path. Indeed, the following proposition holds: Proposition 1. Let (σ,τ be a pair of perutations, S 1,T 1,S 2,T 2,...,S n,t n be its path and (v,h = F n (σ,τ be its transfor. Then one has S i = (v(i,h(i 1 and T i = (v(i,h(i (2 for every i {1,...,n}, where we let h(0 = h(n for the sake of siplicity. Notice that the path goes rightwards (resp. leftwards according to whether v is increasing or decreasing and goes upwards (resp. downwards according to whether h is increasing or decreasing (see Figure 3 for an exaple. Also observe that the functions v and h need not to be perutations; for instance this is the case for the perutation pair of Figure 2 (c. The transfor of the perutation pair of a convex perutoino has special properties, that can be observed in Figure 3 (right. To illustrate the, we introduce the following definition. Definition 1. The pair (v,h Φ n is said to be adissible whenever (v,h S(n S(n and, setting v = v 1 (1, h = h 1 (1, v = v 1 (n, and h = h 1 (n, one has 1 = v h < v h,

5 v is increasing in {1,...,v } and decreasing in {v,...,n}, h is decreasing in {1,...,h }, increasing in {h,...,h }, decreasing in {h,...,n}, with h(n > h(1. The set of adissible pairs in Φ n shall be denoted by Φ A n. This definition is justified by the following fact: Proposition 2. Let (σ,τ Π n be a pair of perutations. Then F n (σ,τ is adissible if and only if the path of (σ, τ is a circuit that can be decoposed into four subpaths directed, in this order, down/rightward, up/rightward, up/leftward, down/leftward. The previous proposition leads us to define adissible a circuit that can be decoposed as in the stateent (see, for exaple, Figure 3 and 4, and to introduce the set Π A n = {(σ,τ Π n the path of (σ,τ is an adissible circuit}. Indeed, the previous proposition can be extended as follows: Proposition 3. The sets Π A n and ΦA n are in bijection via F n. Proof. Proposition 2 iplies that F n (Π A n Φ A n ; we need to prove bijectivity. Given two perutations (v,h with v(1 = 1, set σ(x = h(v 1 (x 1 and τ(x = h(v 1 (x. Letting F n (σ,τ = (v,h, one can easily verify that v(i = v (i and h(i = h (i by induction on i. For uniqueness, it is sufficient to use the definition of F n and Proposition 1. The preiage (σ,τ Π A n of a (v,h Φ A n is called the antitransfor of (v,h, and the path of (σ,τ is called the anticircuit of (v,h. In particular, since convex perutoinoes are adissible circuits, we obtain: Corollary 1. If (σ,τ is the perutation pair of a convex perutoino P, then its transfor (v,h is adissible. Moreover, A = S v, B = T h, C = S v and D = T h. Observe however that, in general, the converse of the previous corollary is not true, because an adissible circuit ay contain crossing points, as shown in Figure 4 (Left. 4 Crossing points At this point, it should be clear that, if (σ,τ Π A n, then either its path is a perutoino or it has crossing points. Such points can ensue only fro one of the following two cases: the up/rightward subpath intersects the down/leftward subpath (crossing point of the first type, as in Figure 4, or the down/rightward subpath intersects the up/leftward subpath (crossing point of the second type, as in Figure 5. Actually, we will show that the crossing points do satisfy stronger conditions. Lea 1. Let (v,h Φ A n sae type. and P be its anticircuit. Then, the crossing points of P (if any are all of the Proof. Let X be a crossing point of first type. Then the down/rightward subpath of P is all included in the square with vertices (1,1 and X; analogously, the up/leftward subpath of P is all included in the square with vertices X and (n,n. This iplies that these subpaths never cross each other, so any other crossing point ust be of the first type.

6 Fig. 3. (Left A convex perutoino P. (Right The diagra of the transfor (v, h (circles represent h, whereas crosses represent v of the perutation pair of P. h(n h(1 h v h Fig. 4. (Left An adissible circuit which is not a perutoino. (Right The diagra of the pair of functions (v,h corresponding to the circuit (circles represent h, whereas crosses represent v.

7 Fig. 5. The anticircuit of (v,h, which is not a perutoino and has crossing points of the second type. Now consider the sequence of crossing points of P, ordered so that their abscissas are (strictly increasing. Clearly, the circuit P passes through all these points once in this order and then again in the reverse order. Notice that also the ordinates turns out to be ordered: if the crossing points are of the first (resp. second type, then they are strictly increasing (resp. decreasing. For the sake of siplicity, thanks to Lea 1, we now focus on adissible pairs whose anticircuit P has only crossing points (if any of the first type. The other case can be dealt with by sietry. Under this hypothesis, the crossing points of P can be classified into two groups, as illustrated in Figure 6. A crossing point X is UL if it is the intersection of an upward segent S µ T µ (where h < µ< v with a leftward segent T λ S λ+1 (where λ > h ; in this case, X = (v(µ,h(λ by Proposition 1. A crossing point X is RD if it is the intersection of a righward segent T ρ S ρ+1 (where h < ρ < v with a downward segent S δ T δ (where δ > h ; in this case X = (v(δ,h(ρ by Proposition 1. T µ S δ S λ+1 X = (v(µ, h(λ T λ T ρ X = (v(δ, h(ρ S ρ+1 S µ T δ (a (b Fig. 6. (a A UL crossing point at indices (µ,λ; (b A RD crossing point at indices (δ,ρ. It is easy to see that the first crossing point of P is UL, the last one is RD, whereas the inner ones alternate. In particular this iplies that the nuber of crossing points is always even. Thus, letting X 1,X 2,...,X 2k 1,X 2k be the ordered sequence of crossing points of P, there exists a sequence of indices µ 1 ρ 1 < µ 2 ρ 2 < < µ k ρ k < δ k λ k < δ k 1 < δ 1 λ 1

8 such that, for every i = 1,...,k, X 2i 1 = (v(µ i,h(λ i and X 2i = (v(δ i,h(ρ i ; note that the crossing points with odd indices are UL whereas those with even indices are RD. Since the abscissa and the ordinates are increasing, we also have and v(µ 1 < v(δ 1 < v(µ 2 < < v(µ k < v(δ k h(λ 1 < h(ρ 1 < h(λ 2 < < h(λ k < h(ρ k. Actually, the points X j s all lay on the diagonal with endpoints (1,1 and (n,n. (On the other hand, if the crossing points are of the second type, it turns out that they all lay on the diagonal with endpoints (1,n and (n,1. Indeed, we show that v(µ i = h(λ i and v(δ i = h(ρ i for every i, that is, the previous chains of inequalities coincide. Consider any UL crossing point X 2i 1 = (v(µ i,h(λ i, and the new circuits (that, in general, ay theselves contain crossing points: S 1,T 1,...,S h,t h,...,s µi,x 2i 1,S λi +1,T λi +1,...,S n,t n X 2i 1,T µi,s µi +1,T µi +1...,S v,t v,...,s h,t h,...,s λi T λi. Observe that, by Corollary 1, the first circuit contains A and B and is included in the square with vertices (1,1 and X 2i 1, whereas the second circuit contains C and D and is included in the square with vertices X 2i 1 and (n,n. Also, by Proposition 1, the second circuit is the anticircuit of the restrictions of v and h to the set {µ i,...,λ i } (up to suitable translations. Hence, such restrictions are bijections onto the sets {v(µ i,...,n} and {h(λ i,...,n}, respectively, and hence one gets v(µ i = h(λ i. Siilarly, any RD crossing point X 2i = (v(δ i,h(ρ i splits the circuit P into two circuits: the one included in the square with endpoints (1,1 and X 2i, and the other included in the square with endpoints X 2i and (n,n. Thus, one obtains v(δ i = h(ρ i for every i = 1,...,k. Hence, the crossing points split the circuit P into 2k + 1 new circuits P j, for j = 0,...,2k (see Figure 7. Each of the has no crossing point, thus it is the boundary of a convex polyoino. Actually, reasoning as above, one can prove that each P j is the boundary of the perutoino whose perutation pair (σ j,τ j is defined as follows (setting δ 0 = 1, µ k+1 = v, and up to suitable traslations of doains and codoains: σ 2i is the restriction of σ to the doain {v(δ i,...,v(µ i+1 } for every i = 0,1,...,k; τ 2i is the restriction of τ to the doain {v(δ i,...,v(µ i+1 }, except for τ 0 (v(µ 1 =v(µ 1, τ 2k (v(δ k = v(δ k, τ 2i (v(δ i = v(δ i for every i = 1,2,...,k, and τ 2i (v(µ i+1 = v(µ i+1, for every i = 0,1,...,k 1; σ 2i 1 is the restriction of σ to the doain {v(µ i,...,v(δ i } for every i = 1,2...,k, except for σ 2i 1 (v(µ i = v(µ i and σ 2i 1 (v(δ i = v(δ i ; τ 2i 1 is the restriction of τ to the doain {v(µ i,...,v(δ i } for every i = 1,2...,k. Intuitively, the pairs (σ j,τ j are the restrictions of (σ,τ to suitable subintervals of {1,...,n}, except for the interval endpoints in correspondence with crossing points. Notice that the perutoinoes with boundaries P 0 and P 2k are both directed-convex, while the other ones are parallelogras. So, we have proved the following theore. Theore 1. Let (v,h Φ A n and P be its anticircuit. Then, either P is the boundary of a convex perutoino, or P has an even nuber 2k of crossing points of the sae type. In the latter case, either all crossing points lay on the diagonal with endpoint (1,1 and (n,n, or they all lay on the diagonal with endpoints (1,n and (n,1. Moreover, P deterines 2k+1 new circuits, each of which is the boundary of a convex perutoino: the 2k 1 inner perutoinoes are parallelogra, whereas the two outer ones are directed-convex.

9 S δ2 P 4 S δ1 S λ2+1 T µ2 = T ρ2 P 3 X 4 X 3 T δ2 = T λ2 S ρ2+1 P 2 T µ1 T ρ1 X 2 S ρ1+1 S µ2 S λ1+1 P 1 P 0 X 1 T δ1 = T λ1 S µ1 Fig. 7. How the crossing points split a circuit into a sequence of perutoinoes. 5 Counting convex perutoinoes Theore 1 provides a precise characterization of the adissible pairs having crossing points. We will call the bad pairs, since they do not define a perutoino. Hence, in order to count the nuber C n of convex perutoinoes of size n, we first obtain the nuber A n of adissible pairs and then the nuber B n of the bad ones. Our ain result is hence given by a subtraction C n = A n B n. To this ai, we recall that in [5] the authors succeeded in giving an explicit forula for counting the nuber of soe subclasses of convex perutoinoes; ore precisely, they proved that the nuber d n of directed-convex perutoinoes with size n is d n = 1 ( 2(n 1, (3 2 n 1 whereas the nuber of parallelogra perutoinoes of size n equals the (n 1-th Catalan nuber p n = c n 1 = 1 ( 2n 2. (4 n n 1 We start by coputing the nuber of adissible pairs: Theore 2. For every n 2, the nuber of adissible pairs is n 2 A n = s t s=0 t=0 u=0 ( n 2 t ( n 2. u+s t Proof. By the definition of adissible pair, we first have to choose the values h, v and h such that 1 h < v h n. Once these values are fixed, take any two subsets of {2,...,n 1}, say V and H, with cardinalities v 2 and h h 1, respectively. Let now v be the unique perutation such

10 that v(1 = 1, v(v = n, v({2,...,v 1} = V, with v increasing in such an interval, and decreasing in the reaining interval {v + 1,...,n}; siilarly, let h be the unique perutation such that h(h = 1, h(h = n, h({h + 1,...,h 1} = H, with h increasing in such an interval, and decreasing (cyclically in the reaining interval {h + 1,...,n} {1,...,h 1}. This is clearly an adissible pair, and it is uniquely deterined by the choice of V and H. So, the nuber of adissible pairs is A n = 1 h <v h n ( ( n 2 n 2 v 2 h. h 1 Substituting s = h 2, t = v 2 and u = v h 1 in the previous suation, we obtain the result. As proved in a separate work [1], the previous suation turns out to be equal to ( 2(n 2 2n4 n 3 (n 2. (5 n 2 As shown in the previous section, bad adissible pairs can be depicted as particular sequences of parallelogra and direct convex perutoinoes. We proceed with two counting leata that will lead to an explicit forula for B n in Theore 3. Lea 2. For every 0, we have k t 1,...,t 2k 1 >0 t 1 + +t 2k 1 =+1 c t1 c t2k 1 = ( 2. Proof. The Catalan nuber c t counts the nuber of trees 1 with 2t edges whose internal nodes have exactly two children. So, the left-hand side of the forula counts the nuber of ordered forests ade by an odd nuber of trees, each being non-trivial and with all internal nodes having two children exactly, where the overall nuber of edges is 2+2. The set F 2+2 of such forests is in bijection with the set T 2+2 of the trees with 2+2 edges, all internal nodes with exactly two children, and the root with an even nuber of children whose half is odd. Indeed, let T 1,...,T 2k 1 be any forest in F 2+2 and consider, for every i, the two subtrees T and T i T 1,T rooted at the two children of the root of T i. The corresponding tree in T 2+2 is obtained by attaching 1,...,T 2k 1,T 2k 1 at a new root. Conversely, every tree in T 2+2 can be obtained fro a suitable forest in F 2+2. Thus, the result is proved if we show that the cardinality of T 2+2 is exactly ( 2. This follows fro the general forula of [3], with R = {2,6,10,14,...}, N = {2} and L = {1}, that yields the generating function T(z = 1+z 2 / 1 4z 2 of the sequence { T }. Since G(z = 1/ 1 4z is the generating function for the central binoial coefficient ( 2, we obtain that Lea 3. For every 0, we have T(z = 1+z 2 G(z 2 = 1+ 0 s=0 ( 2( s 4 s = s 1 Here and henceforth, by tree we ean ordered rooted tree. ( 2 z 2+2. ( 2 (2+1. i

11 Proof. We prove that the generating function for the left-hand side is the sae as the one for the righthand side. The left-hand side is a convolution, whose generating function is the product of 1/(1 4z and 1/ 1 4z, i.e., (1 4z 3/2. For the right-hand side, notice that =0 ( 2 (2+1 z ( 2 = 2 z + =0 = 2z d ( 1 + dz 1 4z =0 ( 2 z = 1 1 4z = (1 4z 3/2. Theore 3. For every n 3, the nuber of adissible pairs that do not define a perutoino is ( 2(n 2 B n = (n 1 4 n 2. n 2 Proof. By Theore 1, if an adissible pair does not define a perutoino, then it defines a sequence of 2k+1 perutoinoes P 0,P 1,...,P 2k where P 0 and P 2k are direct-convex and P 1,...,P 2k 1 are parallelogra. Letting n i be the size of P i, we have i n i = n+2k, since each crossing point X i (i {1,...,2k} coincides with both the upper-rightost corner of P i 1 and lower-leftost corner of P i. Hence the nuber of adissible pairs that do not define a perutoino are n/2 B n = 2 k=1 n 0,...,n 2k 2 n 0 + +n 2k =n+2k d n0 p n1 p n2k 1 d n2k where d n and p n are the nubers of direct-convex and parallelogra perutoinoes of size n, respectively. The factor 2 accounts for the syetry between crossing points of the first and second type. Now, set r = n 0 1, s = r 1+n 2k and t i = n i 1 for every i {1,...,2k 1}. Recalling Equations (3 and (4, we have B n = 2 n 2 s=2 n/2 k=1 t 1,...,t 2k 1 1 t 1 + +t 2k 1 =n 1 s ( 1 c t1 c t2k 1 4 s 1 r=1 ( 2r r Applying Lea 2 with = n 2 s and Equation (1 with n = s, we obtain B n = 1 2 = 1 2 = 1 2 n 2 s=2 n 2 s=0 n 2 s=0 ( 2(n s 2 n s 2 ( 2(n s 2 n s 2 ( 2(n s 2 4 s n s 2 ( 4 s 2 ( 4 s 2 n 2 s=0 ( 2s s ( 2s = s ( 2(n s 2 n s 2 ( 2(n 2 n 2 ( 2s s ( 2(s r s r = ( 2(n 2. n 2 Now, applying Lea 3 with = n 2 to the first suand, and using again Equation (1, we obtain the result. Fro Theores 2 and 3 and equation (5, we are now able to show the ain result. Corollary 2. The nuber of convex perutoinoes of size n 2 is ( 2(n 2 C n = A n B n = 2(n+24 n 3 (2n 3 n 2 The first few ters of the sequences A n, B n and C n are given in Table 1.

12 Table 1. The nuber A n of adissible pairs, the nuber B n of adissible pairs with crossings and the nuber C n of convex perutoinoes of size n. n A n B n C n Conclusions In this paper we have presented a novel technique to study perutoinoes: we defined the transfor of a pair of perutations that can be thought of as a sort of duality. More precisely, even though the set of pairs of perutations (σ, τ defining a convex perutoino is difficult to be described directly, its iage through the transfor F n can be fully characterized. As a consequence, we were able to obtain an explicit forula for the nuber C n of convex perutoinoes (Corollary 2. We point out that a recursive generation technique (naely, the ECO ethod has been independently proposed in [4], where an equivalent forula counting the nuber of convex perutoinoes has been found. We conclude rearking that the generating function of {C n } n is algebraic. Hence, it would be interesting to investigate if there is a natural unabiguous context-free language which is in bijection with the class of convex perutoinoes. Acknowledgeents We would like to thank Sione Rinaldi for introducing us to the proble, and Sebastiano Vigna and Alberto Bertoni for soe useful discussions. References 1. Alberto Bertoni and Roberto Radicioni. A result on soe recurrence relations containing the iniu function. Technical Report , Dipartiento di Scienze dell Inforazione, Università degli Studi di Milano, Marie-Pierre Delest and Gérard Viennot. Algebraic languages and polyoinoes enueration. Theoret. Coput. Sci., 34(1-2: , Eeric Deutsch. Ordered trees with prescribed root degrees, node degrees, and branch lengths. Discrete Math., 282(1-3:89 94, Filippo Disanto, Andrea Frosini, Renzo Pinzani, and Sione Rinaldi. Personal counication. February Irene Fanti, Andrea Frosini, Elisabetta Grazzini, Renzo Pinzani, and Sione Rinaldi. Polyoinoes deterined by perutations. In Fourth Colloquiu on Matheatics and Coputer Science Algoriths, Trees, Cobinatorics and Probabilities, pages , Federico Incitti. Perutation diagras, fixed points and Kazhdan-Lusztig R-polynoials. Technical Report 16, Institut Mittag-Leffler, Federico Incitti. On the cobinatorial invariance of Kazhdan-Lusztig polynoials. Journal of Cobinatorial Theory, Series A, 113(7: , Christian Kassel, Alain Lascoux, and Christophe Reutenauer. The singular locus of a Schubert variety. J. Algebra, 269(1:74 108, Robert Sedgewick and Philippe Flajolet. An introduction to the analysis of algoriths. Addison-Wesley Longan Publishing Co., Inc., Boston, MA, USA, 1996.