Discrete Mathematics and Theoretical Computer Science AC,2003,217–228 Generating functions for the area below some lattice paths
非谓语动词
Donatella Merlini夫人的英文
Dipartimento di Sistemi e Informatica,Via Lombroso6/17,50134,Firenze,Italia
merlini@dsi.unifi.it尼日利亚的首都
1Introduction
In this paper we consider a model of random walks previously studied in[BM02]:each walk starts(at time0)from a point p0of and at time n,one makes a jump x n;the new position is given by the recurrence p n p n1x n where,when p n1k,the x n’s are constrained to belong to afixed t P k(that is,the possible jumps depend on the position of the walk).We call paths on the walks under this model. In combinatorics,it is classical to reprent a particular walk as a path in a two dimensional lattice, thus the drawing corresponds to the walk(of length n)linking the points0p01p1n p n.It is also convenient to reprent all the walks of length n as a tree of height n,where the root(at level0 by convention)is labeled with the starting point of the walks and where the label of each node at level n encodes a possi
ble position of the walk.Figure1illustrates the generating tree reprenting the walks on with jumps P11starting in0(and up to length n4).The branches01010,01012, 01210,01212,0123201234correspond to the well-known Dyck paths of length 4,and are drawn in Figure2.
This kind of trees are known in the literature as generating trees and in the last years have been widely studied.They have been ud for thefirst time,without any specific name,in[CGHK78]and successively this concept can be found in[Wes95,Wes96].Generating trees are a device to reprent the development of many class of combinatorial objects which can then be enumerated by counting the different labels in the various levels of the tree(,[BBMD02,BLPP99]).
The walks on are homogeneous in time,since the t of jumps when one is at altitude k is indepen-dent from the time.When the positions p n’s are constrained to be nonnegative,we talk about paths on (this corresponds to deal with generating trees with positive labels).
When the ts P k’s are equal to afixed t P,the corresponding walks have been deeply studied both in combinatorics and in probability theory(,[BF02]and the included references);in particular, the walks can be generated by context-free grammars(e,e,g,[MRSV99]).
1365–8050c2003Discrete Mathematics and Theoretical Computer Science(DMTCS),Nancy,France
218Donatella
Merlini
Fig.2:Dyck paths of length4and their area.
When the ts P k’s are unbounded,walks are not homogeneous in space,since the t of available jumps depends on the position,and it is not possible to generate them by context-free grammars.However,if the ts P k’s have a“combinatorial”shape,it is reasonable to hope that the generating function associated to the corresponding walk would have some nice properties.In[BM02],veral class of such walks are prented and the nature of the generating fu
nction counting the number of walks of length n going from 0to k(or,equivalently,the number of nodes with label k at level n in the tree)is studied.
In this paper,we examine the walks on related to the concept of proper Riordan arrays and study,in particular,the area below the paths and the x-axis.
The concept of Riordan arrays provides a remarkable characterization of many lower triangular arrays that ari in combinatorics.The theory has been introduced in[SGWW91]and then examined cloly from a theoretical and practical viewpoint in[Spr94,MRSV97].Recently,in[MV00],the connection between proper Riordan arrays and generating trees has been investigated and the resulting trees are called proper generating trees;this relation allows to combine the counting capabilities of both approaches and can be exported in the context of lattice paths.
The area below paths is a combinatorial problem which has some important connections with permu-tations and the internal path length in various types of trees and has been studied in veral contexts(e, e.g.,[BK01,DF93,GJ83,Knu73,MSV96,Sul98,Sul00]).
As it will be shown in Section2,the total area below all paths on of length n is related to the total internal path length,weighted with the values of the labels in the nodes and up to level n,of the corres
ponding generating tree.The internal path length of proper generating trees has been studied in [Mer02];here,we prent similar results in the context of lattice paths thusfinding an explicit generating function for the total area below all the paths under the prent model and,in particular,for tho with an infinite t of jumps.The involved generating functions are expresd in terms of the functions d t h t defining the associated proper Riordan array(e Theorem4).
Generating functions for the area below some lattice paths219 2Background
In this ction we summarize some results on generating trees and Riordan arrays which will be uful in the next ctions.The complete theory of Riordan arrays,the proofs of their properties and the relation with generating trees can be found in[MRSV97,MV00].
2.1Generating trees
ranth
A generating tree is a rooted labeled tree with the property that if v1and v2are any two nodes with the same label then,for each label l,v1and v2have exactly the same number of children with label l.In order to specify a generating tree we have to specify a label for the root and a t of rules explaining how to derive from the label of a parent the labels of all of its children.For example,Figure1illustrates the upper part of the generating tree which corresponds to the following specification:
root:0
rule:k k1k1(1) We can associate a matrix to any generating tree:a matrix associated to a generating tree(AGT matrix, for short)is an infinite matrix d n k n k where d n k is the number of nodes at level n with label k c c being the label of the root.
For example,for rule(1)we have the following AGT matrix:
n k
01
2
0201
4
(3)
1wth t
In the quel we always assume that d00;if we also have h00then the Riordan array is said to be proper;in the proper-ca the diagonal elements d n n are different from zero for all n The most
isaac220Donatella Merlini simple example is the Pascal triangle for which we have
n k t n
1
1t
k
where we recognize the proper Riordan array d t h t11t or d t w11t1w as can be easily proved from(2)and(3).
Proper Riordan arrays can also be defined in terms of two quences A a i i with a00and Z z0z1z2(e,[Rog78,Spr94,MRSV97])such that every element d n1k1can be expresd as a linear combination,with coefficients in A,of the elements in the preceding row,starting from the preceding column:
d n1k1a0d n k a1d n k1a2d n k2
and such that every element in column0can be expresd as a linear combination,with coefficients in Z, of all the elements of the preceding row:
d n10z0d n0z1d n1z2d n2
The generating functions A t and Z t of the quences are related to the pair d t h t by the fol-lowing formulas:
h t A th t(4)
d t
d0
1th t
(6)
and this formula can be ud,in the prent context,to compute the total number of paths of length n
In fact,the following connection between proper Riordan array and generating trees holds:
Theorem2.Let c a j z j j0a00and k c and let
root:c
rule:k c z k c c1a k c c2a k c1k1a0(7) be a generating tree specification.Then,the AGT matrix associated to(7)is a proper Riordan array D defined by the triple d0A Z such that
d01A a0a1a2Z z0z1z2
Generating functions for the area below some lattice paths221 On the contrary,if D is a proper Riordan array defined by the triple d0A Z with d01and a j z j
j0then D is the AGT matrix associated to the generating tree specification(7).
Note we only consider nonnegative labels,thus when a rule gives a negative value,we simply ignore this label.Moreover,the powers in the rule denote repetition of the same label,so we write k r instead of k k k
r
As an application of the previous theorem,let us consider the rule(1),thefirst few applications of which give:
01102213
We thus recognize rule(7)with A10100and Z01000that is,A t1t2and Z t t By applying formulas(4)and(5)wefind that the pair d t h t defining the AGT matrix for the rule(1)corresponds to:
d t h t 1
2t2
Formula(6)in this ca gives the following generating function:
12t
价值观的英文
2t2t1
1t2t23t36t410t520t635t770t8O t9
3The area below proper paths on
In this ction we examine paths on described by the rule(7)with c0The root of a generating trees can have any label and in fact in[Mer02]the internal path length has been studied for a generic label c in the root.In the prent context,since the label of the root reprents the starting point of each path,we can always assume c0:different values of this label correspond to translate each path,along the y-axis, by the same quantity.From here on,we will call proper paths on the paths described by the following rule:
root:0
rule:k0z k1a k2a k1k1a0(8) where,according to Theorem2,A a0a1a2and Z z0z1z2are the A and Z-quences of the associated AGT matrix.
We explicitly obrve that when the generating functions A t and Z t are polynomials,that is,A and Z have afinite number of coefficients different from zero,then the generating tree corresponding to rule (8)defines walks on with afinite t of jumps.More generally,(8)defines walks with an infinite t of jumps,which depends on the position.The powers in the rule can be interpreted as colors that can be ud to distinguish various occurrences of the same jump.
Note,in particular,the jump1is the only positive jump allowed and it always belongs to the t of avail
able jumps since,by hypothesis,a00lausanne
Our interest consists in computing the total area between the paths of length n and the x-axis;this quantity is related to the total internal path length,up to level n,in the corresponding generating tree, weighted with the value of each node label.Referring to Figure1,we have a total path length equal to1 for paths up to level1equal to4for paths up to level2equal to12for paths up to level3and equal to
222Donatella Merlini 34for paths up to level4In fact,we will prove that the generating function counting the total path length for rule(1)is given by:
P t 1t
12t14t2t4t
212t334t484t5212t6488t71162t8O t9
Any path of length n in a proper generating tree can be en as a histogram of length n:in fact any label k in the path can be associated to a column of k cells and by juxtaposing veral columns in such a way that their lowest cells are at the same level,we obtain what we call a histogram.Thus,the total internal path length corresponds to the total area of histograms,if we compute the area as the s
um of the columns height.On the other hand,it is evident that the area of the histograms is strictly related to the area of the regions between the paths and the x-axis,as shown for example in Figure2for Dyck paths of length 4We therefore define the area A W of a path圣诞英语手抄报
W0p01p1n p n
on as the sum of the ordinates of its points:
A W
n ∑
i0
p i
In this paper,we are interested in the generating function P t∑n0P n t n counting the total area P n of all the paths of length n described by rule(8).In particular,we’llfind a formula which only depends on the functions d t and h t defining the associated proper Riordan array.More generally,one can study the j th moment A W j∑n i0p j i of a path W The method we propo in this ction can be ud tofin
d the total moments of any order j for all the paths of length n but the computations in this ca become very complicated.
In order to study the total area of proper paths on we consider the total internal path length of the corresponding generating tree.The total sum of the labels in all the paths from level0to level n in the generating tree can be computed,level by level,by summing the labels counted with their multiplicity;if P i n is the sum of the labels at level i counted with their multiplicity we have:
P n
n ∑
i0
P i n
Figure3illustrates how P i n can be computed:if wefix level n and consider a label r at level i,0i n the multiplicity of this label is given by the number of nodes at level n i in the marked sub-tree,that is,in the generating tree having the same specification(7)but root labeled r.This quantity must be multiplied by the number of nodes at level i having label r and obviously by the value r of the label.O
n the other hand,we know that the element d i r of the associated proper Riordan array counts the number of nodes at level i with label r So,if we let f j t be the generating function counting the number of nodes at a given level in the generating tree having root labeled j we have:
P n
n
∑
i0
adrian lamoP i n
n
∑
i0咒愿
∑
r0
d i r r t n i f r t(9)
Thefirst step in the computation of the sum(9)consists in the computation of the generating function f j t
