Noncrossing Trees and Noncrossing Graphs

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1 Noncossing Tees and Noncossing Gaphs William Y. C. Chen and Shey H. F. Yan Cente fo Combinatoics, LPMC, Nanai Univesity, Tianjin, P.R. China Submitted: Sep 18, 2005; Accepted: Nov 30, 2005; Published: Aug 14, 2006 Mathematics Subject Classifications: 05A05, 05C30 Abstact We give a paity evesing involution on noncossing tees that leads to a combinatoial intepetation of a fomula on noncossing tees and symmetic tenay tees in answe to a poblem poposed by Hough. We use the epesentation of Panholze and Podinge fo noncossing tees and find a coespondence between a class of noncossing tees, called pope noncossing tees, and the set of symmetic tenay tees. The second esult of this pape is a paity evesing involution on connected noncossing gaphs which leads to a elation between the numbe of noncossing tees with n edges and descents and the numbe of connected noncossing gaphs with n + 1 vetices and m edges. 1 Intoduction A noncossing gaph with n vetices is a gaph dawn on n points numbeed in counteclocwise ode on a cicle such that the edges lie entiely within the cicle and do not coss each othe. Noncossing tees have been studied by Deutsch, Feetic and Noy [2], Deutsch and Noy [3], Flajolet and Noy [4], Noy [6], Panholze and Podinge [7]. It is well nown that the numbe of noncossing tees with n edges equals the genealized Catalan numbe c n = 2n+1( 1 3n ) n. In this pape we ae concened with ooted noncossing tees. We assume that 1 is always the oot. A descent is an edge (i, j) such that i>jand i is on the path fom the oot 1 to the vetex j. Atenay tee is eithe a single node, called the oot, o it is a oot associated with thee tenay tees. A symmetic tenay tee is a tenay tee which can be decomposed into a tenay left subtee, a cental symmetic tenay tee and a tenay ight subtee that is a eflection of the left subtee, as shown in Figue 1. Let S n be the set of symmetic tenay tees with n intenal vetices. A noncossing tee is called even if the numbe of descents is even. Othewise, it is called odd. Denote the electonic jounal of combinatoics 13 (2006), #N12 1

2 T R T Figue 1: T is the eflection of T and R is symmetic. by E n and O n the sets of even and odd noncossing tees with n edges, espectively. Let s n,e n,o n be the cadinalities of the sets S n, E n, O n, espectively. Deutsch, Feetic and Noy [2] have shown that ( ) 1 3m if n =2m, 2m +1 m s n = ( ) (1.1) 1 3m +1 if n =2m +1. 2m +1 m +1 Recently, Hough [5] obtained the geneating function fo the numbe of noncossing tees with n edges and a pescibed numbe of descents. He also deived the following elation: e n o n = s n. (1.2) Hough [5] ased the natual question of finding a combinatoial intepetation of the above identity (1.2). In this pape, we obtain a paity evesing involution on noncossing tees that leads to a combinatoial intepetation of (1.2). Ou combinatoial intepetation of (1.2) elies on the epesentation of noncossing tees intoduced by Panholze and Podinge [7]. Given a noncossing tee T,wemay epesent it by a plane tee with each vetex labeled by L o R with the additional equiement that the oot is not labeled, and the childen of the oot ae labeled by R. Such a (L, R)-labeled tee epesentation of T is obtained fom T (as a ooted tee) by the following ule: Given any non-oot vetex j of T, suppose that i is the paent of j. If i>jthen the label of the vetex coesponding to j is labeled by L; othewise, it is labeled by R. These two equivalent epesentations of noncossing tees ae illustated by Figue 2. It is obvious that a descent in the noncossing tee in the fist epesentation coesponds to a L-labeled vetex in the second epesentation. The second esult of this pape is an expession of the numbe of noncossing tees with n edges and descents in tems of the numbe of connected noncossing gaphs with n + 1 vetices and edges. Noncossing gaphs have been extensively studied by Flajolet and Noy [4]. They deived the following fomula fo the numbe of connected noncossing gaphs with n + 1 vetices and edges, that is, N n, = 1 ( )( ) 3n 1. (1.3) n n +1+ n 1 the electonic jounal of combinatoics 13 (2006), #N12 2

3 l l l Figue 2: Two epesentations of a noncossing tee Hough [5] found a combinatoial intepetation of the elation between the descent geneating function of noncossing tees and the geneating function fo connected noncossing gaphs. By using the Lagange invesion fomula he obtained the following explicit fomula fo the numbe of noncossing tees with n edges and descents, d n, = 1 ( )( ) n 1+ 2n. (1.4) n n 1 n +1 As the second esult of this pape, we pesent a paity evesing involution on connected noncossing gaphs and obtain an expession fo the numbe d n, in tems of the numbes N n,m. 2 An involution on noncossing tees In this section, we give a paity evesing involution on noncossing tees which leads to a combinatoial intepetation of the elation (1.2). We use the epesentation of noncossing tees intoduced by Panholze and Podinge [7]. Let T be an even noncossing tee with n edges and v be a non-oot intenal node of T. A vetex v is called a pope vetex if it has an even numbe of left childen but has no ight child. If T is odd, that is, T has an odd numbe of descents, then v is said to be pope if v has an even numbe of ight childen but has no left child. Othewise, v is said to be impope. A noncossing tee is said to be pope if evey non-oot vetex is pope. Othewise, it is said to be impope. It is obvious that each odd noncossing tee is impope. Let us use T n to denote the set of pope noncossing tees with n intenal nodes and let t n denote the cadinality of T n. Let us ecall that a plane tee is said to be an even tee if each vetex has an even numbe of childen. Chen [1] gives a bijection ψ between the set of even plane tees with 2n edges and the set of tenay tees with n intenal nodes. A simila bijection is obtained by Deutsch, Feetic and Noy [2]. Hee we give a bief desciption of this bijection. Suppose that T is an even plane tee with 2n edges. We use the following pocedue to constuct a tenay tee with n intenal vetices. Step 1. Constuct two plane tees T 1 and T 2 based on T : T 1 is the subtee containing the electonic jounal of combinatoics 13 (2006), #N12 3

4 the oot and the fist two subtees of T,wheeasT 2 is the subtee of T obtained by emoving the fist two subtees of the oot. Step 2. Combine T 2 with T 1 by joining T 2 as the last subtee of the oot of T 1. Step 3. Repeat the above pocedue fo all the nontivial subtees (with at least two vetices) of the oot. Since each non-oot vetex of a pope even noncossing tee has only an even numbe of left childen and has no ight child, we can discad the labels of its childen and epesent a pope tee as a plane tee such that each subtee of the oot is an even tee. We define amapσ : T n S n as follows. The map σ: Let T be a pope even noncossing tee. Let T 1 be the fist subtee of the oot. The map is defined by a ecusive pocedue. Step 1. Assign a vetex as the oot and let ψ(t 1 ) be the fist subtee of the oot and its eflection be the thid subtee of the oot. Step 2. Let T 2 be the subtee obtained fom T by deleting T 1,andletσ(T 2 )bethe second subtee of the oot. The above map σ is clealy a bijection between T n and S n. Figue 3 is an example. Figue 3: The map σ Theoem 2.1 The map σ is a bijection between the set of pope noncossing tees with n edges and the set of symmetic tenay tees with n intenal vetices. By using even plane tees as an intemediate stuctue, we may obtain a combinatoial intepetation of (1.2) by constucting an involution on impope noncossing tees which changes the paity of the numbe of descents. Theoem 2.2 Thee is a paity evesing involution on the set of impope noncossing tees with n edges. So we have the following elation e n o n = t n. (2.1) the electonic jounal of combinatoics 13 (2006), #N12 4

5 Poof. Let T be an impope noncossing tee with n edges. Tavese T in peode and let v be the fist encounteed impope node. Define the map φ as follows: Case (1), if T is an odd tee and v has at least one left child, then φ(t ) is obtained by changing its ightmost left child to a ight child and changing all the childen of the non-oot vetices tavesed befoe v to left childen; Case (2), if T is an odd tee and v has no left childen but has an odd numbe of ight childen, then φ(t ) is obtained by changing all the childen of v to left childen and changing all the childen of non-oot nodes tavesed befoe v to left childen. If T is an even tee and v has at least one ight child, then one can evese the constuction in Case (1). If T is an even tee and v has no ight child and has an odd numbe of left childen, then the constuction in Case (2) is also evesible. Hence the map φ is an involution on the set of impope noncossing tees with n edges. Moeove, one sees that this involution changes the paity of the numbe of descents. Thus, we obtain the elation (2.1). An example of the above involution is illustated in Figue 4. l l l l l l l Figue 4: The involution φ Combining the bijections in Theoems 2.1 and 2.2, we get a combinatoial intepetation of the elation (1.2). Note that equation (1.2) leads to the following two combinatoial identities 2m 1 =0 2m =0 ( 1) ( 2m 1+ ( 1) ( 2m + )( ) 4m 2m +1 )( ) 4m +2 2m +2 = = ( ) 2m 3m, 2m +1 m ( ) 3m +1. m +1 3 An involution on connected noncossing gaphs In this section, we aim to establish a connection between the numbe of noncossing tees with n edges and descents and the numbe of connected noncossing gaphs with n +1 vetices and m edges. the electonic jounal of combinatoics 13 (2006), #N12 5

6 Theoem 3.1 We have the following elation 2n 1 ( ) m n ( 1) m n N n,m =( 1) d n,. (3.1) m=n Let G be a connected noncossing gaph with vetex set {1, 2,...,n+1}. We may constuct a unique spanning tee of G, which is called the canonical spanning tee of G. This constuction can be viewed as a efomulation of the tavesal pocedue of Hough [5]. Since G is noncossing, any cycle of G can be epesented by a sequence (i 1,i 2,...,i ) such that i 1 <i 2 < <i,and(i 1,i 2 ), (i 2,i 3 ),...,(i 1,i )and(i,i 1 ) ae the edges of the cycle. Fo a cycle (i 1,i 2,...,i ) epesented in the above fom, we may delete the edge (i 1,i 2 ) to bea the cycle until we obtain a spanning tee. An example is shown in Figue 5. We have the following uniqueness popety of the canonical spanning tee. Poposition 3.2 Let G be a connected noncossing gaph. The canonical spanning tee of G does not depend on the ode of the cycles chosen in the edge deletion pocedue. Poof. Suppose that we get two diffeent canonical spanning tees T and T of a connected noncossing gaph G by using diffeent odes of the cycles fo the edge deletion pocedues. Assume that (i 1,i 2 ) / E(T )and(i 1,i 2 ) E(T ). Suppose that C 1,C 2,,C and C 1,C 2,,C ae the cycles encounteed in the edge deletion pocedues fo T and T. Since (i 1,i 2 ) / E(T ), we may assume that (i 1,i 2 ) E(C j )andi 1 and i 2 ae the minimum and the second minimum numbes of C j. Since (i 1,i 2 ) E(T ), we may find the minimum intege t such that afte beaing the cycle C t by deleting the appopiate edge, the numbes i 1,i 2 ae no longe the minimum and second minimum numbes in the cycles. Let G t be the subgaph of G obtained by the opeations of beaing the cycles C 1,C 2,...,C t 1. Let C =(i 1,i 2,...,i ) be a cycle in G t. Then the cycle C t can be epesented as (i s,i s+1,j 1,j 2,...,j p )o(i 1,i,j 1,j 2,...,j p ). In the fist case, since the gaph G is noncossing, j 1,j 2,...,j p ae on the cycle C. Assume that j p = i q. Afte beaing C t, thee is also a cycle (i 1,i 2,...,i s,i q,...,i )withi 1,i 2 being the minimum and second minimum numbes. In the second case, afte beaing C t, thee is also a cycle (i 1,i 2,...,i,j 1,j 2,...,j p )withi 1,i 2 being the minimum and second minimum numbes. Both the above two cases contadict with the assumption fo C t. Thus T and T ae identical. Convesely, given a noncossing tee T with n edges and a subset S of its descents, we can constuct a connected noncossing gaph by using the bijection of Hough [5] which can be descibed as follows: Fo each descent (i, j) ins, find the maximal path of consecutive descents fom j bac to the oot, and let the fist vetex on this path be v. Fom the neighbos of the vetices on the path fom v to i except fo the vetices on the path, choose the neighbo w as the lagest vetex less than j; Then add the new edge (w, j) to T. We call the new edge (w, j) thecompanion edge of the descent (i, j). An edge in G is said to be fee if it is not in the canonical spanning tee T. A descent (i, j) in the canonical spanning tee of a connected noncossing gaph is said to be satuated the electonic jounal of combinatoics 13 (2006), #N12 6

7 Figue 5: The canonical spanning tee if its companion edge is contained in the connected noncossing gaph. Othewise, it is said to be unsatuated. We now need to conside connected noncossing gaphs in which some of the fee edges ae maed. Denote by N n,m, the set of connected noncossing gaphs with n + 1 vetices and m edges and maed fee edges. It is clea to see that the cadinality of the set N n,m, is given by ( ) m n N n,m. Denote by N n, the set of connected noncossing gaphs with n + 1 vetices and maed fee edges. A descent (i, j) in the canonical spanning tee of a connected noncossing gaph is said to be maed if its companion edge is maed. Denote by D n, the set of connected noncossing gaphs with n + 1 vetices and n + edges such that each descent in its spanning tee is maed. It follows that D n, = d n,. We will be concened with the set N n, D n,, that is, the set of connected noncossing gaphs with n + 1 vetices and maed fee edges which contain at least one unmaed descent. Note that two descents (i, j) and(i,j ) can not shae an end vetex, namely, j j. A descent (i, j) issaidtobesmallethanadescent(i,j )ifj<j. We now give an involution on the set N n, D n, that eveses the paity of the numbe of fee edges. Theoem 3.3 Thee is an involution on the set N n, D n, that eveses the paity of the numbe of fee edges. Poof. Let G be a connected noncossing gaph in N n, D n, with m n fee edges. We define a map ψ as follows. Fist, find the minimum unmaed descent (i, j). We have two cases. Case 1: The descent (i, j) is satuated in G. We delete the companion edge of (i, j) to get a connected noncossing gaph with n + 1 vetices, m n 1fee edges and maed fee edges. Case 2: The descent (i, j) is not satuated in G. Weadd the companion edge of (i, j) to get a connected noncossing gaph with n + 1 vetices, m n + 1 fee edges and maed fee edges. The opeations in the two cases clealy constitute an involution that changes the numbe of fee edges by one. As a consequence of Theoem 3.3, we obtain the identity (3.1). the electonic jounal of combinatoics 13 (2006), #N12 7

8 To conclude this pape, we ema that Theoem 3.1 can be deduced fom the fomulas (1.3) and (1.4) fo N n, and d n, and the following identity 2n 1 ( )( )( ) ( )( ) ( 1) m n 3n m 1 m n n 1+ 2n =, (3.2) n +1+m n 1 n 1 n +1 m=n which can be veified by using the Vandemonde convolution [8, p. 8] ( ) n m = ( )( ) m + i 1 n ( 1) i. i j i+j= Acnowledgments. We would lie to than the efeee fo helpful suggestions. This wo was suppoted by the 973 Poject on Mathematical Mechanization, the National Science Foundation, the Ministy of Education, and the Ministy of Science and Technology of China. Refeences [1] W.Y.C. Chen, A geneal bijective algothm fo inceasing tees, Systems Science and Mathematical Sciences, 12 (1999) [2] E. Deutsch, S. Feetic and M. Noy, Diagonally convex diected polyominoes and even tees: a bijection and elated issues, Discete Math. 256 (2002) [3] E. Deutsch and M. Noy, Statistics on non-cossing tees, Discete Math., 254 (2002) [4] P. Flajolet and M. Noy, Analytic combinatoics of non-cossing configuations, Discete Math., 204 (1999) [5] D.S. Hough, Descents in noncossing tees, Electonic J. Combin., 10 (2003) N13. [6] M. Noy, Enumeation of noncossing tees on a cicle, Discete Math., 180 (1998) [7] A. Panholze and H. Podinge, Bijections fo tenay tees and noncossing tees, Discete Math., 250 (2002) [8] J. Riodan, Combinatoial Identities, John Wiley & Sons, Inc., the electonic jounal of combinatoics 13 (2006), #N12 8