ECO:a methodology for the enumeration of combinatorial objects

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ECO:a methodology for the enumeration ofcombinatorial objectsElena Bracucci a , Alberto Del Lungo a , Elisa Perogola a & Renzo Pinzani aa Dipartimento di Sistemi e Informatica, Via Lombroso 6/17, Firenze, 50134, ItalyVersion of record first published: 29 Mar 2007.

To cite this article: Elena Bracucci , Alberto Del Lungo , Elisa Perogola & Renzo Pinzani (1999): ECO:a methodology for theenumeration of combinatorial objects, Journal of Difference Equations and Applications, 5:4-5, 435-490

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EC0:A Methodology for the Enumeration of Combinatorial Objects

ELENA BARCUCCI, ALBERT0 DEL LUNGO, ELlSA PERGOLA and RENZO PINZANI*

Dipartimento di S i s t e m i e Informatics, Via Lornbroso 6/1Z 50134 Arenze, Italy

(Recewed J u l y 1998; In f ina l form 8 January 1999)

In this paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. The basic idea of this method is the following: by means of an operator that performs a "local expansion" on the objects, we give some recursive constructions of these classes. We use these constructions to deduce some new functional equations verified by classes' generating functions. By solving the functional equations, we enumerate the combinatorial objects according to various parameters. We show some applications of the method referring to some classical combinatorial objects, such as: trees, paths, polyominoes and permutations.

Keywords: Combinatorial enumeration; Trees; Paths; Polyominoes; Permutations

1 INTRODUCTION

In order to enumerate a class of combinatorial objects, it is natural to examine how the objects grow according to a fixed parameter. If we are lucky, this allows us to deduce a recursive construction of the class which can be translated into a functional equation verified by the generating function of the class under consideration. By solving this functional equation, we enumerate the combinatorial objects according to various parameters. We can find this "enumerative philosophy" in many papers and two general methods that use it are: Theory of Species of Structures [lo] and Object Grammars [20].

* Corresponding author. E-mail: pinzani(a]dsi.unifi.it.

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436 E. BARCUCCI et al.

In this paper, we present another general method (called ECO) that uses the same enumerative philosophy. The basic idea of this method is the following: given a class S of combinatorial objects and a parameter p of S, let us consider the set S, = {xES: p(x) = n) ; if we are able to define an operator 6 which satisfies two particular conditions, then we can construct each object YES,,^ from another object XES, and every YES,,^ is obtained from only one XES,. Therefore, we have a recursive construction of S's elements from which we can often deduce a functional equation verified by S's generating function. We can solve these particular functional equations by using Bousquet- Melou lemma [9] or some other similar techniques.

In the sequel, we apply the ECO method to some classes of classical combinatorial objects such as: trees, paths, polyominoes and permutations. We obtain some constructions of the classes from which new functional equations follow, yielding the generating function of these classical objects according to various parameters.

We wish to point out that this method is easily applicable to permutations. Indeed, if S is a class of permutations with length as parameter p, then each permutation TES,,~ is obtained from a permutation ~ E S , by inserting (n-k 1) into d. Some authors [3,13,22,25,35,38,39] have been extensively using this strategy for enumerating many classes of permutations with forbidden subsequences. The purpose of this paper is to show how this idea of "local expansion" can be extended from permutations to other objects to obtaining a general methodology for the enumeration of combinatorial objects.

2 THE ECO METHOD

Let S be a class of combinatorial objects. Let p be a parameter of S (i.e., p : S -+ N') and S, = ( ~ € 5 : p(x) = n ) . An operator 6 on S is a function from S, to 2'"+1, where 2'"+' is the power set of S,,,.

PROPOSITION 2.1 Let 6 be an operator on S . If6 satisfies the following conditions:

1 . for each Y E S,+ there exists X E S, such that YE 6 ( X ) , 2 . let XI, X2€Sn and X I # X2, then 6(Xl) n 6(X2) = 8,

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ECO: METHODOLOGY FOR ENUMERATION 437

then the following family of sets: F,,, = {6(X): VXES,) is a partition

of s,,,. Given a class S of combinatorial objects and a parameter p on S , if we are able to define an operator 6 which satisfies conditions 1 and 2, then Proposition 2.1 allows us to construct each object YES,,^ from an object XES, and every YES,,,, is obtained by only one XES,. Therefore, such an operator on S yields a recursive description of the class S. In some cases, this recursive description allows us to deduce a functional equation verified by the generating function of S. We can solve the functional equation verified by the generating function of S. We can solve the functional equations arising from our constructions by using the following lemma [9] or some other similar analytical tools. We often denote any functionfls, x, y , q ) by A s ) for the sake of brevity.

LEMMA 2.2 (Bousquet-MClou) Let R = R[[s , x , y , q]] he the algebra of the formal power series in variables s, x, y and q having real coefficients, and let A be a sub-algebra of R such that the series converge for s = 1 . Let A(s, x, y , q ) be a formal power series in A. Let us assume that:

where e(s) ,As) and g(s) are some given power series in A. Then:

where

By introducing the concept of generating tree, we can also use the method to determine some bijections between classes of con~binatorial objects. If we assume that there is only one element of minimal size in S , then we can describe its recursive construction by using a rooted tree in which the vertices on the nth level represent the elements of S,. The root of the tree is the smallest element, and an element YES,,, is

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the child of the element XES, if Y belong to d(X) (see Fig. 3) . We call this tree the generating tree of S related to the operator 19. If S and St are two different classes of combinatorial objects and their generating trees are isomorphic, then there is a bijection between S and S t . This technique has been widely used in [22,25,38] on the permutations with forbidden subsequences. We wish to point out that the generating trees have been used for the first time in [13] for enumerating Baxter permutations.

We can elegantly describe the recursive constructions (i.e. the structure of the generating trees) by means of the notion of rewriting rule. We label each vertex of the generating tree with a sequence of infor- mations from which we can deduce the number of its sons. We use the following notation to connect the label of a parent with the label of its k sons:

We call this function rewriting rule. The label (p) includes some information from which we obtain the value k. Sometimes one rewriting rule is sufficient to determine the generating tree's structure. In the following sections, we will show that the generating trees of many classical combinatorial objects are characterized by only one rewriting rule, and furthermore the label p is equal to k. There are some other cases in which one rewriting rule is not sufficient. For these constructions, we introduce labels leading to a system of rewriting rules that completely determine the generating tree's structure. For instance, a recursive construction of the permutations sortable by two passages through one stack is characterized by a complex system of rewriting rules (see [19]).

We wish to point out that, by defining an operator I9 over a class S , we get a system RR of rewriting rules characterizing the construction of S , obtained by means of 8. This system RR describes some other recursive constructions of combinatorial objects' classes which are in bijection with S . We call this set of classes associated to RR the combinatorial space of RR.

In the following sections, we apply the ECO method to some classes of combinatorial objects enumerated by classical sequences of numbers. We obtain some recursive constructions of the objects

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characterized by some systems RR of rewriting rules, and we use them to enumerate the objects according to various parameters. Moreover, we show that objects enumerated by the same numbers are described by the same system RR of rewriting rules, belonging therefore to the same combinatorial space.

3 CATALAN COMBINATORIAL SPACE

The Catalan numbers:

are well-known integers that arise in many combinatorial problems. Gould's research bibliography [23] contains over 500 references per- taining to Catalan numbers and for an exposition of their basic combinatorial properties see Hilton and Pederson [27].

We apply the ECO method to some classes of combinatorial objects enumerated by Catalan numbers: plane trees, Dyck paths, parallelogram polyominoes and Catalan permutations. The unitary elements of these objects are: nodes for the trees, steps for the paths, cells for the parallelogram polyominoes and positive integers for the Catalan permuations. The first step of the method consists in defining the operator 19. Since 19 produces an object YES,,, from another object X E S , , we have that Y is obtained by inserting a unitary element in some sites of X. Therefore, we have to determine the sites of objects in S,: a site has the property that the insertion of a unitary element in one of these sites gives an object of S,,,,. The next step amounts to select a set of sites in each object such that the insertion of the unitary elements in each of them defines a construction for the class of combinatorial objects we are referring to. At this stage the operator 19 satisfying conditions 1 and 2 of Proposition 2.1 is determined; the selected sites are said active sites. We will show that each class we take into consideration can be constructed by means of an operator 19 characterized by the following rewriting rule:

( k ) -+ ( 2 ) ( 3 ) . . . (k) ( k + I ) .

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Let A be a finite set. A plane tree (in the sense used by Knuth [29, p. 3051) is an ordered partition { { a } ; A l ; A Z ; . . . ;A,} of A, such that a € A, and each Ai is a plane tree. The elements of A are called internal nodes (or nodes) and the node a is called the root of the plane tree. A plane tree is an unlabelled tree (i.e., its nodes are indistinguishable). The sets A l , A2, . . . , A, are the subtrees of the root. Each node of a plane tree can be the root of some subtrees contained in the tree. The number of a node's subtrees is called the degree of the node. An internal node of degree zero is called a leaf. Each root is said to be the jather of its subtrees' roots, which, in turn, are called sons of their father. The sons of the same father are referred to as brothers. The level of a node is defined as follows: the root's level is 0, and all the other nodes' level is one unit higher than their father's. A plane tree's kth level is the set of its nodes at level k. Let P be a plane tree. If a and a' are nodes of P, we say that { a l , a2,. . . , a,} is a path of length n from a to a' if a = a l , a' =a, and ak is the father of ak+, for 0 < k < n. The internal path length of P is the sum (evaluated over all the internal nodes) of the lengths of the paths from the root to each internal node. The right branch length of P is the length of the path from its root to its rightmost internal node (the last one in the preorder traversal).

3.1.1 The Operator

Let P be the class of plane trees. The parameter p : P + Nf is the number of internal nodes. Therefore P, is the set of the plane trees having n internal nodes. Let us extend each plane tree by attaching a special son to each of its internal nodes so that the special node become the rightmost son of the internal node. These new nodes are called external nodes (see Fig. I). If PEP,, then we obtain a plane tree P ' E P , , ~ by replacing any of P's external nodes by an internal one. Therefore, the external modes are the sites of the plane tree.

Let us now define the operator 19 and consequently the set of active sites of a plane tree. A traversal of a plane tree is a linear arrangement of its nodes. The preorder traversul is the traversal obtained by visiting the root and the subtrees of the roots from left to right (see Fig. 1).

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ECO: METHODOLOGY FOR ENUMERATION 44 1

Levels

internal node 0 external node

FIGURE 1 A plane tree and its preorder traversal

DEFINITION 3.1 Let PEP,. W e denote:

x l (P) as P's last external node in the preorder traversal, 3 ( P ) as the set of P's external nodes that follow x l (P) in the preorder traversal (the sites in 3 ( P ) are the active sites of P).

Then 6 ( P ) is the set of the plane trees obtained from P by replacing a node of F ( P ) with an internal node.

In Fig. 2 is shown the set of the plane trees obtained by performing 29 on a plane tree P . It is easy to prove that:

PROPOSITION 3.1 The operator 29 on the class of plane trees satisfies conditions 1 and 2 o f Proposition 2.1.

Therefore, by means of 29 we can construct P,+, from P,,. This recursive construction of plane trees is illustrated by the generating tree of Fig. 3. The root of the generating tree is the plane tree made up of only one node.

We now determine the rewriting rule characterizing the generating tree of P obtained by means of z9. Let P E P . The right branch length of P is equal to the number of P's active sites (i.e. 3 ( P ) ' s nodes). Therefore, if P's right branch length is k, then 8 ( P ) is made up of k

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FIGURE 2 The set 8(P) of a plane tree obtained by means of preorder traversal.

FIGURE 3 The generating tree of the plane trees obtained by means of 19.

plane trees: PI, PZ, . . . , P k l , Pk. They are all sons of P in the generating tree. Moreover, from 6's definition it follows that the number of active sites of these k trees is 2 , 3 , . . . , k, k + 1, respectively (the reader can check this property on the generating tree in Fig. 3). Consequently

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PROPOSITION 3.2 The generating tree of plane trees is isomorphic to the tree having its root labelled (1) and recursively defined by the following rewriting rule:

(k) --t (2) (3) . . . (k) (k + 1).

We wish to point out that, if we take another hierarchical traversal (i.e. each node precedes its sons and its right brother), then we get another operator 29 on the class P that satisfies conditions 1 and 2 of Proposition 2.1. For instance, the level traversal (i.e, the visiting of the root and the nodes on increasing levels from left to right) is a hierarchical traversal, and by using it we obtain another recursive construction of plane trees described by the same rewriting rule. Moreover, the same strategy can also be applied to many other sub- classes of plane trees (see [2]).

3.1.2 Plane Trees' Enumeration

Many authors [18,24,28,29,36] have been studying plane trees enumeration by using different tools. We find some results obtained by these authors and we deduce a new generating function for plane trees according to the number of their internal nodes, the number of their leaves, their right branch length and their internal path length.

We begin by translating the construction obtained by 19 into an equation. Let PEP, we denote the following parameters on P (see Fig. 4): 0 r(P), the right branch length,

n(P), the internal nodes number, 0 1(P), the leaves number, 0 i(P), the internal path length.

P's generating function according to the above-listed parameters is :

We often denote A(s, x, y, q) by A(s). P's right branch length is equal to the number of 3(P)'s nodes (i.e., I3(P) I = r(P)). Furthermore, for each k ~ [ l , . . . , r(P)], there is e€F(P ) , such that e's level is k (see Fig. 4).

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E. BARCUCCI et al.

LEVELS

L - - - - - - - _ I

FIGURE 4 Parameters r(P) , n(P), l (P )

5

and i(P) of a plane tree.

FIGURE 5 The construction obtained by 79.

Let F ( P ) = {el, e2,. . . , e , ( p ) } , where ek is the node at level k. We now apply the operator 6 on P. Each tree of 6(P) is obtained from P by replacing a node of 3 ( P ) by an internal node (see Fig. 5).

If we replace ek , with k € [1, . . . , r(P) - 11, then we obtain P 1 ~ 6 ( P ) , such that: r(P1) = k + 1 , n(P1) = n(P) + 1 , I(P1) = I(P) + 1 and i(P1) =

i (P)+k+ 1;

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if we replace e r ( ~ ) , then we obtain P 1 ~ 1 9 ( P ) such that: r(P1)= r(P) + 1, n(P1) = n(P) + 1 , l (P1) = I(P) and i(P1) = i(P) + r(P) + 1.

Therefore, the translation of this construction into the generating function A(s, x, y, q) gives us:

The operator 8 applied to the set P gives all the plane trees P such that n(P) > 1 , and consequently:

PROPOSITION 3.3 The plane trees' generating function A(s, x, y, q) verifies the following functional equation:

By means of Lemma 2.2 and Proposition 3.3, we get the following:

THEOREM 3.4 The generating function A(s, x , y , q) is given by:

with

and

xi~Snq~~(n+5)/2 "-1 Eo ( s ) = 1 - s2xyq2 1 k+ 1

n ( l y s q 1, ,>O ('4; q)n+~ k=0

where we denote (a; q ) , = ny~i ( 1 - a d )

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Let us now take the generating function A(l ,x , y , l ) into consideration. This generating function is algebraic of degree 2 (see for example [24, p. 3851) and can be obtained from the functional Eq. (3.1.1). By setting q = 1 in Eq. (3.1.1) it follows that:

( 1 - s(1 + x - xy) + s 2 x ) ~ ( s , x , y , 1 ) = sxy(1 - s + sA(1, x , y, 1 ) ) . (3.1.2)

If 1 - s(1 + x - xy) + s2x = 0, then 1 - s + sA(1, x , y, 1 ) = 0. The solution of this algebraic equation is:

therefore:

By putting y = 1, we get:

that is the well-known generating function of Catalan numbers. Consequently, we have the following classical properties related to Catalan and Narayana numbers (see for instance [28,30]).

COROLLARY 3.5 The number of plane trees having n + 1 internul nodes is the nth Catalan number and the number of plane trees having n + 1 internal nodes and k leaves i.~

The Eq. (3.1.2) gives us:

A ( ~ , & . Y , 1 ) = SXY ( 1 - s + sA( I , x , y , 1 ) ) .

1 - s - sx + sxy + s2x

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ECO: METHODOLOGY FOR ENUMERATION

By setting y= 1 we obtain:

from which the property related to Delannoy numbers follows (see

[111).

COROLLARY 3.6 The number of plane trees with n internal nodes and right-branch length k is:

The last remark concerns the generating function A(1, x, q, 1). We denote this function as A(x,q) for short. From Theorem 3.4 we deduce that:

where:

It is easy to prove the two following identities:

Therefore:

Eo (xq, q) Eo (.x, q) + El (-%/, 4 ) A(s , q) = .uq = xq

Eo (x, 9) Eo (x3 q)

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This equation gives the following classical result:

that is, the generating function of plane trees according to the number of their internal nodes and their internal path length is the continued fraction studies by Ramanujan (see [21,24]).

3.2 Dyck Paths

A path is a sequence of points in N x N, a step of a path is a pair of two consecutive points in the path. A Dyck path is a path P= (so, s1, . . . , sZn) such that so = (O,O), ~2~ = (2n, 0), only having northeast (si = (x, y), s i + ~ = (x + 1, y + 1)) or southeast (si = (x, y), (si+, = (x + 1, y - 1)) steps (see Fig. 6). The number of northeast steps is equal to the number of southeast steps. The path's length is the number of its steps. A peak (resp. valley) is a point si such that the step (sipl, si) is a northeast (resp. southeast) step and the step (si,si+l) is a southeast (resp. northeast) step. The height h(si) of a point si is its ordinate. The extremity s i + ~ of a northeast step (si, s ~ + ~ ) is a non-decreasingpoint. The southeast steps' last sequence of a Dyck path is called last descent.

The number of Dyck paths having length 2n is the nth Catalan number (see [12]).

height

4

FIGURE 6 A Dyck path.

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3.2.1 The Operator

Let P be the class of Dyck paths. The parameter p : P + N+ is the number of northeast steps (i.e. the semilength) and P , is the set of Dyck paths having length 2n. If PEP,, then we obtain a Dyck path P'EP,,, by inserting a peak in any point of P. This means that the points of a path are its sites.

DEFINITION 3.2 If PEP, , then d (P) is the set of Dyck paths obtained ,from P by inserting a peak in any point of P's last descent; these points are the active sites of P.

The set of Dyck paths obtained by performing 29 on a Dyck path P is shown in Fig. 7. The operator d constructs P,+l from P , and it satisfies conditions 1 and 2 of Proposition 2.1. This recursive construction of Dyck paths is illustrated by the generating tree of Fig. 8. The root of the generating tree is the Dyck path made up of only two steps.

We now determine the rewriting rule that characterizes the generating tree of P obtained by means of d . Let P E P , if P's last descent length is k (i.e. last descent contains k points), then the number of P's active sites is k. Therefore, d(P) consists of k Dyck paths:

FIGURE 7 The set of paths obtained by means of operator 19

FIGURE 8 The generating tree of the Dyck paths obtained by means of 19.

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PI , PZ, . . . , Pk-1, P k They are all sons of P in the generating tree. Moreover, from 6's definition it follows that the number of active sites of these k paths is: 2 ,3, . . . , k, k + 1 , respectively. Consequently:

PROPOSITION 3.7 The generating tree of Dyck paths is isomorphic to the tree having (2) as its root label and whose nodes labels are recursively defined by:

( k ) --t ( 2 ) (3) . . . ( k ) ( k + 1).

The recursive construction of Dyck paths and plane trees is characterized by the same rewriting rule; the bijection between these two classes determines some relations between paths and trees parameters.

Let P E P ; the area of P is the sum of P's non-decreasing points' heights. We denote the following parameters on P (see Fig. 6):

0 d(P), the last descent length, 0 l(P), the semilength, 0 p(P), the peaks number, 0 a(P), the area.

If we number the points of P's last descent from right to left in increasing order (see Fig. 9), we have:

if 6 inserts a peak at the last descent's kth point, with k~ [l, . . . , d(p) - 11, then we obtain P1 ~.19(P), such that: d(P1) = k + 1, I(P1) = I(P) + 1, p(P1) =p(P) + 1, and a(P1) = a(P) + k , if 29 inserts a peak at the last descent's d(P)th point, then we obtain PIE 6(P) such that: d(P1) = d(P) + 1, I(P1) = I(P) + 1 ,p(P1) =p(P) and a(P1) = a(P) + d(P).


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TABLE I Correspondence between plane trees and Dyck paths parameters

Plane trees Dyck paths - - - - - -

R~ght branch length Last descent length Internal nodes number Semilength + 1 Leaves number Peaks number Internal path length Area + sem~length + I

FIGURE 10 A parallelogram polyomino.

From these relations and the recursive construction of the plane trees it follows that, if P is the plane tree corresponding to P, then:

r (P) = d(P) , n (P) = I(P) + 1, 1(P) = p ( P ) , and i (P) = a(P)+ I(P) + 1.

Consequently, we deduce the relations between trees and paths parameters in Table I. The enumerative results obtained for plane trees can be translated on the Dyck paths by using the relations listed in Table I.

3.3 Parallelogram Polyominoes

Let us consider the plane n= Z x Z . A cell is a unit square in n, a polyomino is a finite connected union of cells having no cut point; polyominoes are defined up to a translation. A column (row) of a polyomino is the intersection between the polyomino and an infinite vertical strip of cells. The area of a polyomino is its number of cells, the perimeter is the number of edges of its border, while its height and width are the numbers of its rows and columns, respectively. Parallel- ogram polyominoes or skew Ferrers diagrams are defined by two paths that only intersect at their origin and extremity and only have North and East steps (see Fig. 10). The semiperimeter of a parallelogram

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polyomino is equal to the sum of its height and width. These polyominoes have been extensively studied and are counted by Catalan numbers according to their perimeter [15]. Their generating function according to the area, width and height is related to q-Bessel functions [9,14,31].

3.3.7 The Operator

Let P be the class of parallelogram polyominoes; the parameter p : P+ N+ is the semiperimeter, Pn is the set of parallelogram polyominoes having perimeter 2n. If PEP,, then we obtain a parallelogram polyomino P ' E Pn+l by inserting a column or a row in P.

DEFINITION 3.3 I f PEP, and l(P) is the length of its rightmost column, then 6(P) is the set of polyominoes obtained from P:

by gluing a column of length k < I(P) to P's rightmost column, 0 by gluing a cell onto the top of P's rightmost column.

The set of parallelogram polyominoes obtained by performing 6 on a paralleogram polyomino P is shown in Fig. 11. The operator 6 constructs Pn+l from Pn and satisfies conditions 1 and 2 of Proposition 2.1. This recursive construction of parallelogram polyominoes is illustrated by the generating tree of Fig. 12. The root of the generating tree is the parallelogram polyomino made up of only one cell.

We now determine the rewriting rule that characterizes the generating tree of P obtained by means of 6. Let P E P ; if the length of P's rightmost column is k - 1, then 6(P) consists of k parallelogram polyominoes: P I , PZ, . . . , Pk-1, Pk. Again, they are all sons of P in the generating tree. Moreover, from 6's definition it follows that the length of the rightmost column of these k polyominoes is 1,2, . . . , k - 1, k, respectively. Consequently:

P

FIGURE 11 The set of parallelogram polyominoes obtained by means of 8.

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FIGURE 12 The generating tree of the parallelogram polyominoes obtained by means of 19.

PROPOSITION 3.8 The generating tree of parallelogram polyominoes is isomorphic to the tree having (2) as its root label and whose nodes labels are recursively defined by:

(k) --t (2)(3) . . . (k)(k + 1). The recursive construction of parallelogram polyominoes and plane

trees is characterized by the same rewriting rule and so there is a bijection between these two classes; it involves some relations between trees and polyominoes parameters.

Let P E P ; we denote the following parameters on P (see Fig. 10):

l (P), the rightmost column length, p(P), the semiperimeter, h(P) , the width, a(P), the area.

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454 E. BARCUCCI et a1

We apply the operator 19 over P:

if 19 adds a column of length k 5 1(P) to P's rightmost column, then we obtain P1€19(P), such that: I(P1) = k, p(P1) =p(P) + 1, h(P1) =

h(P) + 1, and a(P1) = a(P) + k; if 19 adds a cell onto the top of P's rightmost column, we obtain P 1 ~ 1 9 ( P ) , such that: l(P1) = I(P) + 1, p(P1) =p(P) + 1, h(P1) = h(P), and a(P1) = a(P) + 1.

From these relations and the recursive construction of the plane trees it follows that, if P is the plane tree corresponding to P, then:

r (P) = I(P) + 1, n ( ~ ) = p ( P ) , and l (P) = h(P) .

Consequently, we deduce the relations between trees and polyominoes parameters in the Table 11. The enumerative results obtained for plane trees can be translated on the parallelogram polyominoes by using the relations listed in Table 11.

We wish to point out that we get classical bijection between Dyck paths and parallelogram polyominoes in which: the peaks of the paths correspond to the columns of the polyominoes and peak's height is equal to column's length (see [8,16]). Moreover, by proceeding as for the plane trees, we can deduce the following well-known result related to q-Bessel functions [14]:

PROPOSITION 3.9 The generating function P(x, t , q) for parallelogram polyominoes according to their width, perimeter and area is:

TABLE I1 Correspondences between plane trees and parallelogram polyominoes parameters

Plane trees Parallelogram polyominoes

Right branch length Rightmost column length + 1 Internal nodes number Semiperimeter Leaves number Width

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where

3.4 Catalan Permutations

A permutation 7r = 7r(l)7r(2) . . . ~ ( n ) on [n] = (1 ,2 , . . . , n ) is a bijection between [n] and [n]. Let S, be the set of permutations on [n].

DEFINITION 3.4 A permutation T E S , contains a subsequence of type r E Sk iff a sequence of indexes 1 5 i( ') < i(*) < . . . < i(k) 5 n exists such that r ( i l ) < 7r(iZ) < . . . < 7r(ik). We denote the set of permutations of S,, not containing subsequences o f type r by S,,(T).

A site for a permutation T E S , is a position lying between two consecutive elements ~ ( i ) and 7r(i+ 1 ) for i ~ [ l , n - 11, or to the left of ~ ( l ) , or to the right of 7r(n). For each T , an active site of a permutation T E S ~ , ( T ) is such that the insertion of ( n + 1) in that site gives a permutation belonging to the set Snil(r) .

Let TES, . The pair (i,j), i < j, is an inversion if ~ ( i ) > d j ) . An element ~ ( i ) is a right minimum if ~ ( i ) < ~ ( j ) , VJE [i + 1 , n].

Consider now S,(321). This class has been studied in [22,29,38] and its cardinality is equal to the nth Catalan number. In [22] is showed that:

PROPOSITION 3.10 Let 7r be a permutation of S,(32l). I f s is an active site of n , then each site an its right is also active.

If n ~ S ~ ( 3 2 1 ) , then d(7r) is the set of pern~utations obtained from 7r by inserting (nt I ) in any active site. The operator 29 on the S1,(321) satisfies conditions 1 and 2 of Proposition 2.1. Moreover, the generating tree of S,(321) is isomorphic to the tree having its root label (2)

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FIGURE 13 The generating tree of S(321) obtained by means of 19.

and recursively defined by the following rewriting rule:

( k ) --t (2) (3) . . . ( k ) ( k + 1).

This recursive construction of S(321) is illustrated by the generating tree in Fig. 13.

For r E S(32 1 ) we denote by:

a(r), the active sites number, n ( ~ ) , the length, m(r), the right minima number,

0 i ( ~ ) , the inversion number.

The active sites are numbered from right to left in increasing order,

The operator 6 is applied on S(321).

0 If 6 inserts (n + 1) into the first active site of .ir, then r1€z9(P) is such that a(nt) = a(r) + 1, n(nl) = n(r) -t 1 , m(r ' ) = m(r) + 1, and i (r I ) = i(.ir).

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TABLE I11 Correspondences between plane trees and Catalan permutations parameters

Plane trees Cataian permutations

Right branch length Active sites number Internal nodes number Length + 1

If 6 inserts (n + 1) into the kth active site of r, with 2 5 k 5 a ( ~ ) , then r 1 ~ 2 9 ( P ) is such that a(nl) = k, n(r1) = n(r ) + 1, m ( r l ) = m ( ~ ) , and i ( r l ) = i ( r ) + k - 1.

From these relations and the recursive construction of the plane trees it follows that, if P is the plane tree corresponding to r, then:

Consequently, we deduce the correspondences given in Table 111. From plane trees' construction we deduce that, the number of T'S

right minima corresponds to the number of internal nodes of P which are not leaves. The number of plane trees having n + 1 internal nodes and k internal nodes which are not leaves is again the Narayana number, and so.

PROPOSITION 3.11 The number o f S,(321)'s permutations having k right minima is

Moreover, from parallelogram polyominoes' construction it follows that if P is the parallelogram polyomino corresponding to T , then

where a(P) , p(P) and h(P) are P's area, semiperimeter and width, respectively. Therefore, from Proposition 3.9 we deduce that the generating function for S(321) according to their length and inversion number is related to q-Bessel functions.

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PROPOSITION 3.12 The generating function for S(321) according to the length and inversion number is:

where

4 MOTZKIN COMBlNATORlAL SPACE

The Motzkin numbers:

arise in a variety of combinatorial situations some of which are described in [Is]. Furthermore, there are some nice algebraic relations between these numbers and some other quantities, such as Catalan numbers and trinomial coefficients [6,17].

We apply the ECO method to some classes of combinatorial objects enumerated by the Motzkin numbers: right-leafed trees, Motzkin paths, steep-parallelogram polyominoes and Motzkin permutations. As for the structure enumerated by the Catalan numbers, we define an operator 19 satisfying conditions 1 and 2 of Proposition 2.1 for the above-mentioned classes of objects and we show that the rewriting rule characterizing their growth, according to a given parameter, is:

( k ) --t ( 1 ) ( 2 ) . . . ( k - l ) ( k + 1).

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4.1 Right-leafed Trees

A right-leafed tree is a plane tree in which each node is the father of one leaf at most and the leaf must be its rightmost son. Each internal node of degree 1 is not the father of any external node (see Fig. 14). Let C be the class of right-leafed trees. From the defintion of right- leafed tree it follows that we can attach an external node to an internal one of a tree P E L if and only if it is not the father of a leaf. Pro- ceeding in the same way as for plane trees we obtain that the set of active sites F ( P ) for P is the set of P's external nodes that follow the last internal node in the preorder traversal (see Fig. 14). The operator 19 is the one defined for the plane trees and satisfies conditions 1 and 2 of Proposition 2.1. The generating tree of this recursive construction of right-leafed trees is illustrated in Fig. 15. We now determine the rewriting rule characterizing the generating tree of ,C obtained by means of .IP. Let P E L . If P's right branch length is k + 1 , then the number of P's active sites (i.e. 3(P) 's nodes) is k. In this casee, d(P) consists of k right-leafed trees: P I , PZ, . . . , Pk. As before, they are all sons of P in the generating tree. Moreover, from 6's definition it follows that the number of active sites of these k trees is: 1,2, . . . , k - 1, k + I, respectively (the reader can check this property on the generating tree in Fig. 15). Consequently:

PROPOSITION 4.1 The generating tree of right-leafed trees is isomorphic io the tree having its root labelled (1) and recursively defined by the following rewriting rule:

(k) --t (1)(2). . . (k - l ) (k + 1).

FIGURE 14 A right-leafed tree.

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FIGURE 15 The generating tree of the right-leafed trees obtained by means of 8.

4.1.1 Right-leafed Trees' Enumeration

The generating function for right-leafed trees according to the number of internal nodes, leaves, right branch length and internal path length is given by:

The right branch length of P E L is equal to the number of 3(P)'s nodes plus one (i.e., r(P) = (3(P)I + 1). Furthermore, for each k E { 1,2, . . . , r(P) - 2, r(P)), there is an e ~ 3 ( P ) of level k. Let 3 ( P ) = {el, ez, . . . , e , . (~ ) -~ , er(P)), where ek is the node at level k. If we apply 19 on P (see Fig. 16), by proceeding as for the class P, we have:

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PROPOSITION 4.2 The right-leafed trees' generating function L(s, x, y , q ) satisfies:

By means of Lemma 2.2 and Proposition 4.2, we obtain the following:

THEOREM 4.3 The generating function L(s, x, y , q ) is given by:

with

and

y q 2 n "-1 2 2(k+l)- E l ( s ) = s 2 ~ 2 y q 3 n ( ~ q * + ' - q Y ) ?

n20 ( ~ q > q)n k=o

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Consider now the generating function L(1, x, y, 1 ) . By setting q= 1 , in Eq. (4.1.3) it follows that:

If 1 + xy - s(l + x ) + s2x = 0, then x(1 - s) - L(1, x, y, 1) = 0. The solution of this algebraic equation is:

therefore:

By putting y = 1, we get:

and we obtain the same result as Donaghey and Shapiro [ I 81. That is,

COROLLARY 4.4 The number of right-leafed trees with n internal nodes is equal to the ( n - 2)th Motzkin number. Moreover, the number of right-leafed trees having n internal nodes and k leaves is

We notice that summing up these numbers over k we get:

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The generating function for right-leafed trees according to the number of internal nodes which are not leaves is:

This function is the generating function for Schroder numbers (see Section 5) and so:

COROLLARY 4.5 The number of right-leafed trees having n internal nodes with degree > 0 is equal to the (n - 1)th Schroder numbers:

Eq. (4.1.4) gives us:

By setting y = 1 we obtain:

and so:

COROLLARY 4.6 The rrumher of right-leafed trees with n internal nodes and right-bvanch length k + 1 is:

The last remark concerns the generating function L(1, x,q, denote this function as L(x ,q ) for short. From Theorem

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deduce that:

where:

and so we obtain:

from which we deduce the following nice q-identity:

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FIGURE 17 A Motzkin path.

4.2 Motzkin Paths

A Motzkin path is a path P = ( s o , s l , . . . ,s,) such that so=(O,O), s, = (n, 0 ) , only having northeast (si = ( x , y) , sf+, = ( x + 1, y + 1 ) ) or southeast (si = ( x , y) , si+l = ( x + 1 , y - 1)) or east (si= (x, y), si+, =

( x + 1, y ) ) steps (see Fig. 17). The last sequence of southeast and east steps of a Motzkin path is called last descent. The number of Motzkin paths of length n is equal to nth Motzkin number [18] and their enumeration according to area and major index gives some nice q-analogs of Motzkin numbers [1,7,21].

4.2.1 The Operator

Let C , be the class of Motzkin paths of length n. The active sites of a Motzkin path are the east steps belonging to its last descent.

DEFINITION 4.1 I f PEP,, then d ( P ) is the set of Motzkin paths obtained,from P:

by replacing an east step of P's last descent with a northeast step and by inserting a southeast step at the end of P,

0 by adding an east step at the end of P.

The set of Motzkin paths obtained by performing t9 on a Motzkin path P is shown in Fig. 18. The operator t9 constructs C,+l from L, and satisfies conditions 1 and 2 of Proposition 2.1. The recursive construction of Motzkin paths is illustrated by the generating tree of Fig. 19.

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FIGURE 18 The set of Motzkin paths obtained by means of operator 19.

FIGURE 19 The generating tree of the Motzkin paths obtained by means of 19.

We now determine the rewriting rule that characterizes this generating tree. Let P E L , if its last descent contains k - 1 east steps, then 8(P) is made up of k Motzkin paths P I , P2,. . . , Pk-,, Pk. From 8's definition it follows that the number of east steps in the last descent of these k paths is: 0, 1, . . . , k - 2, k , respectively. Consequently:

PROPOSITION 4.8 The generating tree of Motzkin paths is isomorphic to the tree having ( 1 ) as its root label and whose nodes labels are recursively defined by:

The recursive construction of Motzkin paths and right-leafed trees is characterized by the same rewriting rule and so we have a bijection between these two classes. By proceeding as for the objects belonging to Catalan combinatorial space, we obtain the following relations between trees and paths parameters, as given in Table IV.

Consider now the area of Motzkin paths. The area is defined as the sum of the heights of paths' non-decreasing points, where a non- decreasing point is the extremity si+l of a northeast or east step

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TABLE IV Correspondences between right-leafed trees and Motzkin paths parameters

Right-leafed trees Motzkin paths

Right branch length Number of east steps in the last descent + 2

Internal nodes number Length + 2 Leaves number Northeast steps number + 1

(si,si+,). For instance, the area of Motzkin path in Fig. 17 is 21. Let P be a Motzkin path. We denote its area by a(P). We number the east steps of its last descent from right to left in increasing order. If I9 replace the kth east step of the last descent, with k ~ [ l , . . . , d(p)] , then we obtain P 1 e 6 ( P ) , such that a(P1) = a(P) + k. On the other hand, if I9 adds an east step at the end of P, we obtain P 1 ~ 1 9 ( P ) , such that: a(P1) = a(P) (see Fig. 18). By proceeding as for the right-leafed trees, we can deduce that the generating function for Motzkin paths according to area is a ratio between two q-Bessel functions:

PROPOSITION 4.9 The generating functions S ( x , t , q) for Motzkin paths according to their number of northeast steps, length and area is:

where

In [ I ] , by means of a different recursive construction, it is shown that:

THEOREM 4.10 The generating function for Motzkin paths according to their number of northeast steps, length and area is given bj':

with

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468

and

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From the previous proposition and theorem it follows that:

4.3 Steep Parallelogram Polyominoes

Let us call south border and north border the two paths defining a parallelogram polyomino. A steep parallelogram polyomino is a parallelogram polyomino whose south border has no pair of subsequent horizontal steps (see Fig. 20). In [5] it is proved that the number of steep parallelogram polyominoes having perimeter 2n + 2 is the nth Motzkin number and their generating function according to the area is related to q-Bessel functions.

4.3.1 The Operator

Let P E L , be the class of steep parallelogram polyominoes having perimeter 2n and I(P) be the length of its rightmost column.

€F FIGURE 20 A steep parallelogram polyomino.

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DEFINITION 4.2 For PEL,, i?(P) is the set of polyominoes obtained from P:

0 by gluing a column of length k < I(P) to P's rightmost column, 0 by gluing a cell onto the top of P's rightmost column.

We note that in this case we do not add a column of length I(P) to P's rightmost column. The operator I9 constructs C,,, from C,, and it is easy to prove that 19 satisfies conditions 1 and 2 of Proposition 2.1. The recursive construction of steep parallelogram polyominoes is sketched in the generating tree of Fig. 21. We now determine the rewriting rule that characterizes this generating tree. Let P E P ; if the length of P's rightmost column is k, then 8(P) consists of k parallelogram polyominoes P I , P2, . . . , PkPl , Pk. Moreover, from 19's definition it follows that the length of the rightmost column of these k polyominoes is 1 ,2, . . . , k - 1, k + 1, respectively. Therefore,

PROPOSITION 4.12 The generating tree of steep parallelogram polyominoes has (1) as its root label and its nodes' labels are recursively

FIGURE 21 The generating tree of the steep parallelogram polyominoes.

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defined by:

( k ) - ( 1 ) ( 2 ) . . . ( k - l ) ( k + 1 ) .

We have a bijection between right-leafed trees and steep parallelogram polyominoes and the relations between the parameters are given in Table V.

The height of a steep parallelogram polyomino corresponds to the number of internal nodes which are not leaves in a right-leafed tree. Therefore, from Corollary 4.5 we get:

COROLLARY 4.13 The number of steep parallelogram polyominoes of height n is equal to the (n - 1)th Schroder number.

From Motzkin paths' construction it follows that, if P is the Motzkin path corresponding to a parallelogram polyomino P, then

p(P) = n ( ~ ) + 2, h (P) = n e ( ~ ) + 1

a ( P ) = a(P) + n ( ~ ) - n e ( ~ ) + 1,

where n ( ~ ) , ne(P) and a ( P ) are its length, number of east steps and area, while p(P), h(P) and a(P) are P's semiperimeter, width and area, respectively. Therefore, from Proposition 4.9 we deduce that:

PROPOSITION 4.14 The generating function P(x, t , q) for steep parallelogram polyominoes according to the width, perimeter and area is:

J l ( x , t , 9) P(x , t , q ) = x t2qqxq- I , tq, q ) = - Jo (x, t , q ) '

where

4.4 Motzkin Permutations

In this section, we consider a class of permutations with forbidden subsequences enumerated by Motzkin numbers [3,22,39]. We introduce the concept of a barred permutation.

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TABLE V Correspondences between right-leafed trees and steep parallelogram polyominoes parameters

Right-leafed tress Steep parallelogram polyominoes

Right branch length Rightmost column length + 1 Internal nodes number Semiperimeter Leaves number Width

DEFINITION 4.3 A barredpermutation ? of [k] is a permutation of Sk having a bar over one of its elements. Let I- be a permutation on [k] identical to ? but unbarred, and .i the permutation of [k - 11 made up of 7's unbarred k - 1 element, rearranged to be a permutation on [k - 11.

Example I f7=41352on[5] , w e h a v e ~ = 4 1 3 5 2 a n d . i = 3142

DEFINITION 4.4 A permutation T ~ S , contains a type T subsequence iff n contains a type .i subsequence that, in turn, is not a type T subsequence. W e denote by s , ( ~ ) the set of permutations of S, containing no type 7 subsequence.

Example Let T = 41352. The permutation ~ = 6 1 4 5 7 3 2 belongs to S7(7) because all its subsequence of type 3142: ~ ( l ) , ~ ( 2 ) , n(5), ~ ( 6 ) = 6173, and ~ ( l ) , 7r(2), ~ ( 5 ) , 747) = 6172 are subsequent of ~ ( l ) , 7r(2), n(3), n(5), ~ ( 6 ) = 61473 and ~ ( l ) , 7r(2), 7r(3), 7r(5), n(7) = 61472, which are of type T = 41352.

Let us now take S,,(321,3142) into consideration. This class has been studied in [3,22,39] and its cardinality is equal to the nth Motzkin number. In [22] it is showed that:

PROPOSITION 4.15 Let 7r be a permutation of Sn(321, 3142). I f s is an active site of T, then each site on its right is also active.

If T E Sn(321, 3142), then 6 ( ~ ) is the set of permutations obtained from n by inserting (n + 1) in any active site. The operator .r9 on the Sn(321, 3142) satisfies conditions 1 and 2 of Proposition 2.1. More- over, the generating tree of S,(321, 3142) has its root labelled (2) and is recursively defined by the following rewriting rule:

(k) - (1)(2). . . (k - l ) (k + 1).

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This recursive constructions of Sn(321, 3142) is illustrated by the generating tree in Fig. 22.

The bijection between right-leafed trees and Sn(321, 3142) permutations and the relations among the parameters are given in Table VI.

The number of right minima in a Motzkin permutation is equal to the number of internal nodes which are not leaves in the corresponding right-leafed tree minus one. Therefore, from Corollary 4.5 we get:

COROLLARY 4.16 The number of S(321,3142) permutations having n right minima is equal to the nth Schroder number.

From Motzkin paths' construction it follows that, if P is the Motzkin path corresponding to a Motzkin permutation n, then

where: n ( ~ ) are a(P) are P'S length and area, while n ( ~ ) amd i ( ~ ) are T'S length and inversion number, respectively. Hence, from Proposi- tion 4.9 we deduce that:

FIGURE 22 The generating tree of S,,(321,3142) obtained by means of $.

TABLE VI Correspondences between right-leafed trees and Motzkin pemutations parameters

Right-leafed trees Motzkin permutations

Right branch length Active sites number Internal nodes number Length + 2

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PROPOSITION 4.17 The generating function M(t, q) for S(321,3142) according to their length and inversion number is:

where

5 SCHRODER COMBINATORIAL SPACE

The Schroder number:

is a classical sequence of integer numbers [34] and there are many classes of combinatorial objects enumerated by it [26,33]. Richard Stanley has recently narrated the fascinating story of this sequence

WI. We apply the ECO method to some classes of combinatorial

objects enumerated by the Schroder numbers: coloured plane trees, Schroder paths, 2-coloured parallelogram polyominoes and Schroder permutations. As for the structures enumerated by the Catalan and Motzkin numbers, we define an operator 6 satisfying conditions 1 and 2 of Proposition 2.1 for the above-mentioned classes of objects and we show that the rewriting rule characterizing their growth, according to a given parameters, is

(k) --, (3) (4) . . . (k) (k + 1) (k + 1)

5.1 2-Coloured Plane Trees

A 2-coloured plane tree is a plane tree whose internal nodes (with degree greater than 0) can be coloured black or white, while the leaves are always black (see Fig. 23).

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black internal node

o white

o external node i (P) = 38

sb lo&

FIGURE 23 A 2-coloured plane tree P.

FIGURE 24 The recursive construction defined by the operator 8 on 2-coloured plane trees.

Let C be the class of 2-coloured plane trees. The set 3 ( P ) of P's active sites is determined by the preorder traversal. The operator 6 replaces each P,EF(P) with a black internal node, or gives the white colour to the last internal node of the right branch and adds a son to it (see Fig. 24).

We now determine the rewriting rule characterizing the generating tree of C obtained by means of 19. Let PEC; if P's right branch length is (k - 1) then 3 ( P ) = {e l , . . . , ek_l) and ei's level is i. 6(P) consists of k 2-coloured plane trees: P1, PZ, . . . , Pk-l, Pk that are all sons of P in the generating tree. Moreover, from 6's definition, it follows that the right branch length of these k trees is 2,3,. . . , k, k, respectively

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FIGURE 25 The generating tree of 2-coloured plane trees obtained by means of 19.

(the reader can check this property on the generating tree of Fig. 25.) Consequently:

PROPOSITION 5.1 The generating tree of 2-colouredplane trees has its root labelled (2) and is recursively defined by the ,following rewjriting rule:

(k) ( 3 ) . . . (k)(k + l)(k + 1).

5.2 2-Coloured Plane Trees' Enumeration

The internal path length of PEC can be partitioned into the internal path length of the white internal nodes ( i , (P)) , the leaves ( i2(P)) and

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the black internal nodes with degree greater than 0 (i3(P)), while o(P) counts the number of white internal nodes (see Fig. 23). The translation of the construction into the generating function follows the same considerations carried out for plane trees. The 2-coloured plane trees's generating function according to the right branch length, the internal nodes number, the leaves number, the white internal nodes number and the internal path length is given in the following proposition:

PROPOSITION 5.2 The 2-colouredplane trees'generating function:

satisfies :

If we set ql = q 2 = q3 = q in Eq, (5.2.5) then the generating function refers to the "total" internal path length of 2-coloured plane trees. We denote C(s, x, y, z , q, q, q) by C(s, x, y, z, q) and we solve this particular case by means of Lemma 2.2, obtaining the following result:

THEOREM 5.3 The generating function C(s, x, y, z, q) is given by:

with

and

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Let us now take the generating function C(l,x,y,z, l , l , 1)= C(1, x, y, z) into consideration; by setting q1 = q2 = q3 = 1 in Eq. (5.2.5), it follows that:

If1 -(I +x+xz-xy)s+x( l +z)s2=0,then(l -s)+sC(l,x,y)=O. The solution of this algebraic equation is

therefore:

By putting y = z = 1, we get:

that is the well-known generating function of Schroder numbers. From (5.2.6) we also have:

and setting y = z = 1 we obtain:

so that we obtain:

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COROLLARY 5.4 The number of 2-colouredplane trees with n internal nodes and right branch length k is:

Consider now the generating function C(x, q) which is the generating function of 2-coloured plane trees according to the number of internal nodes and internal path length. From Theorem 5.3 we deduce that:

where:

It is easy to prove the two following identities:

therefore:

and

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from which we deduce the following q-identity:

- - 2x9 - xq.

1 +xq2 - 2xq2

1 +xq3 - 2xq4 1 +xq4- . . .

5.3 Motzkin Paths

The paths studied in Section 4.2 are enumerated by the Motzkin numbers according to their length; if we enumerate the same paths according to the northeast and east steps number then we obtain the Schroder numbers. We notice that this is equivalent to enumerate Schroder paths as defined in [7] according to their length. Let P be a Mctzkin path, the active sites of P arc dcfincd as the poifii~ Gdongilig to its last descent. Figure 17 shows a path with 8 active sites and 11 northeast and east steps. The operator 29 inserts a northeast step in each active sites of P and a southeast step at the end of P and also add an east step at the end of P (see Fig. 26). In [4] it is proved that this operator satisfies conditions 1 and 2 of Proposition 2.1. The generating tree of this recursive construction is illustrated in Fig. 27. We now determine the rewriting rule that characterizes this generating tree. We number the active sites of a Motzkin path P from right to left in increasing order. If P has n northeast and east steps and contains k points in the last descent, then 29(P) consists of (k + 1) paths, with (n + 1) northeast and east steps PI, P 2 , . . . , Pk, Pk+l It follows that the number of points in the last descent of these ( k + 1) paths is 2, . . . ,(k + l), (k + I), respectively and consequently:

icl . . p-'f-, FIGURE 26 The recursive construction obtained by means of tY on Motzkin paths.

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FIGURE 27 The generating tree for the Motzkin paths enumerated according to the northeast and east steps number.

PROPOSITION 5.5 The generating tree of Motzkin paths enumerated according to the number of northeast and east steps has its root labelled (2) and is recursively defined by the following rewriting rule:

Propositions 5.1, 5.5 say that 2-coloured plane trees and Motzkin paths grow in the same way according to the internal nodes number and the northeast and east steps number, respectively. This means that these two classes of combinatorial objects are in bijection. By proceeding as for the objects in Catalan numbers' combinatorial space, we obtain the relations between trees and paths parameters as given in Table VII.

The correspondences illustrated in Table VII say that the generating functions of Motzkin paths enumerated according to the last descent length, northeast and east step number, east step number and area is: D

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TABLE VII Correspondences between 2-coloured plane trees and Motzkin paths parameters

2-coloured plane trees Motzkin paths

Right branch length Number of points in the last descent Internal nodes number - 1 Northeast and east steps number Internal path length of black internal nodes - Area internal nodes number

White internal nodes East steps 2 x Internal nodes number - Length white nodes number - 2

The functional equation obtained from Eq. (5.2.5) can be solved following the same steps described for Catalan structures, so we have:

PROPOSITION 5.6 The generating function of Motzkin paths enumerated according to the northeast and east step number, east step number and area is:

where:

In [4], by means of a different recursive construction, it is shown that:

THEOREM 5.7 The generating function Cpath(l, X, y, q ) = EI (x, Y , q ) / Eo(x, y, q ) where:

therefore the q-identity of Corollary 5.8,follows.

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5.4 2-Coloured Parallelogram Polyominoes

A 2-coloured parallelogram polyomino is a parallelogram polyomino whose columns can be coloured grey or white (see Fig. 28). This class has been studied in [4] and its cardinality, according to the semiperimeter, is equal to the nth Schroder number. Let P be a 2-coloured parallelogram polyomino and r(P) be its uppermost row length. A construction for this class of combinatorial objects is defined by the operator 'L9 that works in the following way: 6(P) is the set of polyominoes obtained by gluing both a row of length i < r(P) onto the P's uppermost row or one white or grey cell onto the right of its uppermost row (see Fig. 29). Consequently:

PROPOSITION 5.9 The generating tree of 2-coloured parallelogram polyominoes enumerated according to their semiperimeter has its root labelled (2) and is recursively defined by the following rew'riting rule:

(k) --, ( 3 ) * * . ( k ) ( k + l ) ( k + 1).

The recursive constructions for 2-coloured plane trees and 2- coloured parallelogram polyominoes are characterized by the same

FIGURE 28 A 2-coloured parallelogram polyomino P.

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FIGURE 29 The recursive construction defined by the operator d on 2-coloured parallelogram polyominoes.

TABLE VIII Correspondences between 2-coloured plane trees and 2-coloured parallelogram polyominoes parameters

- --

2-coloured plane trees 2-coloureciparallelogram polyominoes

Right branch length - 1 Uppermost row length Internal nodes number Semiperimeter Leaves Rows Internal path length of leaves - leaves number Area White internal nodes Grey columns

rewriting rule and so we have a bijection between these two classes. The correspondences between trees and polyorninoes parameters follow immediately by looking at the parameters changes that follow the application of the construction in both structures (see Table VIII).

The correspondences illlustrated in Table VIII say that the generating function of 2-coloured parallelogram polyominoes (Fig. 30) enumerated according to the uppermost row length, columns number,

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FIGURE 30 The generating tree of 2-coloured parallelogram polyominoes.

row's number and area is:

The functional equation obtained from Eq. (5.2.5) can be solved following the same steps described for Catalan structures, so we have:

PROPOSITION 5.10 The generating function of 2-coloured parallelogram polyominoes enumerated according to the columnS number, row's number and area is: D

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where:

In [4], by means of a different recursive construction, it is shown that:

THEOREM 5.11 The generating function Cpol(1, X, y, q ) = EI (x, Y , 911 Eo(x, y , q) where:

Proposition 5.10 and Theorem 5.11 give us:

5.5 Schroder Permutations

Take now S, (4231,4132) into consideration. This class has been studies in [4,39] and its cardinality is equal to the nth Schroder number.

As for the Catalan permutations, if s is an active site of 7r6Sn(4231,4132) then all sites on its right are also active (see Fig. 31). The operator 29 inserts ( n + 1) into each active site of T (see Fig. 32) and satisfies conditions 1 and 2 of Proposition 2.1. The generating tree of this recursive construction for Schroder permutations is sketched in Fig. 33.

It is easy to prove [4] the following:

PORPOSITION 5.13 The generating tree of Schroder permutations has its root labelled (2) and is recursively defined by the following rewriting

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FIGURE 3 1 A permutation a € S(423 1,4132).

FIGURE 32 The operator 29 on a~S(4231,4132).

rule:

(k) --t (3) . . . (k)(k + l ) (k + 1).

Following the way carried out for the objects belonging to the Catalan combinatorial space, we apply the constructions described for 2-coloured plane trees and S(4231,4132) permutations respectively, we fully develop the parameters changes in both cases so we deduce the relations between trees and permutations parameters as given in Table IX.

Remark 5.1 follows immediately.

Remark 5.1 From Corollary 5.4 we have that the number of Schroder permutations with length n and k active sites is:

k - 2 n - k

n - 1 i=o

The correspondences in the Table IX say that the generating function of Schroder permutations enumerated according to the active site

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ECO: METHODOLOGY FOR ENUMERATION

FIGURE 33 The generating tree of Schroder permutations.

TABLE IX Correspondences between 2-coloured plane trees and Schroder permutations parameters

2-colouredplane trees Schroder permutations

Right branch length Active sites number - 1 Iiiteiiia: nodes i l~~iibei- LcngLh Internal path length of black internal nodes - Inversions number internal nodes number

White internal nodes + 1 Right minima

number, length, right minima number and inversions number is:

The functional equation obtained from Eq. (5.2.5) can be solved following the steps described for Catalan structures, so we have:

PROPOSITION 5.14 The generating function qf Schroder permutations enumerated according to the length, right minima number and inversions number is: D

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6 CONCLUSIONS

We described a unifying method (called the ECO method) for enumerating some classes of combinational objects. It has been applied on classical combinatiorial objects such as plane trees, lattice paths, polyominoes and permutations, revealing not only its simplicity but also its powerfulness. Indeed the classical generating functions are easily obtained and new identities appeared. The ECO method can be applied not only to objects whose generating function is algebraic like the examples above, but also be to objects whose generating functior! is not algebraic such as, for instance, ~ ( 4 i 3 2 ) permutations [32].

References

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ECO:a methodology for the enumeration of combinatorial objects

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