Forest Volume Decompositions and Abel-Cayley-Hurwitz Multinomial

Forest volume decomp ositions and

Ab el-Cayley-Hurwitz multinomial expansions

Jim Pitman

Technical Rep ort No. 591

Department of Statistics

University of California

367 Evans Hall 3860

Berkeley,CA94720-3860

March 29, 2001; revised Octob er 24, 2001

Abstract

This pap er presents a systematic approach to the discovery,inter-

pretation and veri cation of various extensions of Hurwitz's multino-

mial identities, involving p olynomials de ned bysumsover all subsets

of a nite set. The identities are interpreted as decomp ositions of for-

est volumes de ned bytheenumerator p olynomials of sets of ro oted

lab eled forests. These decomp ositions involve the following basic for-

est volume formula, which is a re nementofCayley's multinomial

expansion: for R S the p olynomial enumerating out-degrees of ver-

tices of ro oted forests lab eled by S whose set of ro ots is R, with edges

directed away from the ro ots, is

P P

jS jjRj1

x : x

s r

s2S r 2R

Supp orted in part by N.S.F. Grants DMS 97-03961 and DMS-0071448 1

1 Intro duction

Hurwitz [21 ] discovered a numb er of remarkable identities of p olynomials in

a nitenumber of variables x ;s 2 S ,involving sums of pro ducts over all

j j

2 subsets A of some xed set S ,witheach pro duct formed from the

subset sums

x := x and x = x x , where A := A. 1

A s A

s2A

For instance, for S =f0; 1g,withx substituted for x and y for x to

0 1

ease the notation, there are the following:

Hurwitz identities [21, I I', I I I, IV]

jAj1 jAj j j

xx + x y + x =x + y + x 2

jAj1 jAj1 j j1

xx + x y y + x =x + y x + y + x 3

A j j

X X Y

jAj jA j jB j

x + x y + x = x + y + x jB j! x 4

A b

b2B

A j j B j j

For j j = n and x 1, by summing rst over A with jAj = k ,theseHurwitz

sums reduce to corresp onding Abel sums [1, 48 ]

; k + nk +

A x; y := x + k y + n k 5

k =0

for particular integers and . Esp ecially, 2 with x w and y = z nw

reduces to Abel's binomial theorem [1]

k 1 nk n

xx + kw z kw =x + z

k =0

and 4 for x w reduces to a classical identity due to Cauchy. Strehl

[56] explains how Hurwitz was led to suchidentities via the combinatorial

problem, which arose in the theory of Riemann surfaces [20 ], of counting

the number of ways a given p ermutation can b e written as a pro duct of a

minimal numb er of transp ositions which generate the full symmetric group. 2

This pap er develops a systematic approach to the discovery,interpre-

tation and veri cation of identities of Hurwitz typ e such as 2-4 . This

metho d is related to, but not the same, as Knuth's metho d in [32] and [31,

Ex. 2.3.4.30]. There Hurwitz's binomial theorem 3 was proved byinter-

preting b oth sides for p ositiveinteger x; y and x ;s 2 asanumber of rooted

forests with x + y + x + j j vertices, sub ject to constraints dep ending on

the x . The present metho d is closer to the approachofFrancon[18], who de-

rived 2 and 3 using a form of Cayley's multinomial expansion over trees,

expressed in terms of enumerator p olynomials for mappings from S to S .

Section 2 intro duces the notion of forest volumes, de ned by the enumerator

p olynomials of sets of ro oted lab eled forests, with emphasis on a general-

ization of Cayley's multinomial expansion over trees, called here the forest

volume formula. This formula relates Hurwitz typ e identities to decomp osi-

tions of forest volumes. Probabilistic interpretations of forest volumes were

developed in [40, 41, 42, 43]. This pap er, written mostly in combinatorial

rather than probabilistic language, is a condensed version of [40 ]. Acom-

bined version of [40] and [41 ] app ears in the companion pap er [44]. Section

3 o ers two di erent extensions of the forest volume formula. Then Section

4 presents a numb er of Hurwitz typ e identities, with pro ofs by forest volume

decomp ositions. Finally, Section 5 p oints out some sp ecializations of these

Hurwitz identities which give combinatorial interpretations of Ab el sums.

2 Forest volumes

Ro oted lab eled forests. Francon's approach to Hurwitz identities is sim-

pli ed byworking exclusively with various subsets of the set

F := fall ro oted forests F lab eled by S g

whose enumerativecombinatorics has b een extensively studied [37 , 2.4 and

3.5], [54 , 5.3]. Each F 2 F is a directedgraph with vertex set S , that is a

subset of S S , each of whose comp onents is a tree with some ro ot r 2 S ,

with the convention in this pap er that the edges of F are directed away from

the ro ots of its trees. The set of vertices of the tree comp onentof F ro oted

at r is the set of all s 2 S such that there is a directedpath from r to s in F ,

denoted r ; s, meaning either r = s or there is a sequence of one or more

F F F F

! s means s ;s 2 F . Let F := fx : s ! xg edges r ! ! s, where s

2 1 2 s 1 3

denote the set of children of s in F . So jF j is the out-degree of vertex

s in the forest F . Note that the F are p ossibly empty disjoint sets with

[ F = S ro ots F , where ro ots F is the set of ro ot vertices F . So the

s2S s

total numb er of edges of F is jF j = jS jjro ots F j and jro ots F j is

s2S

the numb er of tree comp onents of F .

Forest volumes. For B F ,theenumerator p olynomial in variables

x ;s 2 S

X Y

jF j

V [F 2 B ]:=V [F 2 B ]x ;s 2 S := x 6

S S s

F 2B s2S

is called here the volume of B , to emphasise that B ! V [F 2 B ]isa

measure on subsets B of F , for each xedchoice of x ;s 2 S withx 0.

S s s

As explained later in this section, this notion of forest volumes includes b oth

the probabilistic interpretations develop ed in [43, 41 , 42 , 40 ], and the forest

volume of a graph de ned by Kelmans [29, 26 , 27].

Consider the volume of all forests F 2 F with a given set of ro ots R.

This volume decomp oses according to the set A = [ F of children of

r 2R r

all ro ot vertices of F . With := S R,thisgives the recursivevolume

decomp osition

jAj

V [ro otsF =R]= x V [ro otsF =A] 7

where each side is a p olynomial in variables x ;s 2 S ,and S = [ R with

R \ =;. Consideration of 7 for small jS j leads quickly to the following

generalization of Cayley's multinomial expansion over trees [13, 47 ]:

Theorem 1 The forest volume formula [13, 45, 47, 18, 11, 40]

For R S , the volume of forests labeledbyS whose set of roots is R is

jS jjRj1

V [ro otsF =R]=x x : 8

S R

Pro of. Observe rst that 8 transforms 7 into the following Hurwitz typ e

identity of p olynomials in variables x and x ;s 2 :

R s

j jjAj1 jAj

j j1

x x : 9 x x + x = x

A R R

But 9 is very easily veri ed directly, and 8 follows from 7 and 9 by

induction on jS j. 2 4

History and alternative pro ofs of the forest volume formula. Cayley

[13] formulated the sp ecial case of 8 with R = fr g, call it the tree volume

formula, along with the sp ecial case of 8 with general R and x 1;s 2 S ,

that is the enumeration

jS jjRj1

jfF 2 F with ro ots F =Rgj = jRjjS j 10

which for jRj = 1 yields the Borchardt-Cayley formula n for the number

of unro oted trees lab eled by a set of n vertices. These sp ecial cases of the

forest volume formula are among the b est known results in enumerativecom-

binatorics. See for instance [45, 47 , 43 , 42, 54 , 7 ] for various pro ofs of the

tree volume formula and [19, 36 , 37 , 57 , 51 , 3] for 10 . The preceding pro of

of the forest volume formula parallels a pro of of 10 by induction, using the

consequence of 7 that the numb er in 10 is jS j; jRj with the recursion

k + n; k = k n; a: 11

a=1

Mo on [37 , p. 33] attributes this pro of of 10 to Gob el [19]. The forest

volume formula can also b e derived by the metho d of Prufer co des [45 ],

which has b een applied to obtain a host of other results in the same vein

[30],[37, Chapter 2]. Another approach to the forest volume formula is to

combine any of the pro ofs of the tree volume formula cited ab ove with the

following:

Reduction of forest volumes to tree volumes. Fix some arbitrary

r 2 R, and decomp ose the volume V [ro ots F =R] according to the tree

T =treeF derived by identifying all ro ot vertices of F with r . So T is

lab eled by fr g[ S R, and the sets of children of T are T := [ F and

r s2R s

T := F for s 2 S R.For each p ossible tree T , it is easily seen that

s s

jT j

V [ro ots F =R; treeF = T ]= x x :

s2S R

Now sum over all T and apply the tree volume formula to deduce the forest

volume formula.

Reformulation in terms of mappings. The forest volume formula is

easily recast as a formula for the enumerator of mappings from S to S whose 5

set of cyclic p oints is a set of xed p oints equal to R. This mapping enu-

merator was derived byFrancon [18, Prop. 3.1 and pro of of Prop. 3.5] from

the Foata-Fuchs enco ding of mappings [17] for jRj2. The corresp onding

enumerator for general R can b e inferred from the discussion in [18, p. 337]

and derived the same way. As indicated in [40 , 7 ], a probabilistic form of

the forest volume formula, expressed in terms of random mappings, can b e

read from Burtin [11]. The general form 8 of the forest volume formula was

given in [40, Th. 1] with a roundab out pro of via random mappings.

Variations of the forest volume formula. For eachvector c ;s 2 S

c = jS jjRj, the identityofcoecients of of non-negativeintegers with

x in 8 reads

c jS jjRj1!

jfF 2 F : jF j = c for all s 2 S gj = : 12

S s s

c !

s2S

In the tree case jRj = 1, this is easily veri ed by a direct argument [43, x5.1].

in the formula The same argumentgives the identity of co ecients of x

for the volume of all forests with exactly k tree comp onents [42, x2]:

jS j1

jS jk

: 13 V [F with k trees]= x

k 1

This in turn is the identity of co ecients of x in the tree volume formula

applied to S [f0g instead of S for some 0 2= S . See Stanley [54, Th. 5.3.4]

for two other pro ofs of 13 . It is obvious that 8 implies 13 by summation

over R with jRj = k , but less obvious how to recover 8 from 13. Neither

do es it seem easy to check the co ecientidentities 12 directly for general

R. By summation of 13 over k , the total volume of all forests lab eled by S

jS j1

V [F 2 F ] = 1 + x : 14

S S S

Hurwitz's multinomial formula. An argument suggested byFrancon[18 ,

p. 337] in terms of mapping enumerators can b e simpli ed as follows. The

forest volume V ro ots F = R can b e decomp osed by classifying F accord-

ing to the sets B of non-ro ot vertices of the tree comp onents of F ,asr

ranges over R. For given B ;r 2 R, there is an obvious factorization of

r 6

the forest volume over disjoint tree comp onents, and the tree volume formula

can b e applied within each comp onent. Therefore

X Y

jB j1

15 V [ro otsF =R]= x x + x

S r r B

r 2R

jS jjRj

where the sum is over the jRj p ossible choices of disjoint, p ossibly empty

sets B ;r 2 Rwith[ B = S R. The equalityofright hand sides of

r r 2R r

8 and 15 is Hurwitz's multinomial formula [21 , VI]:

X Y

jB j1 j j1

16 x x + x = x x + x

R R r r B

r 2R

whose binomial case jRj = 2 is 3. As a check on this circle of results,

16 is easily derived by induction on jRj from its binomial case, whichwas

interpreted combinatorially byKnuth [32] and [31 , Ex. 2.3.4.30]. The forest

volume formula can then b e read from the consequence 15 of the tree volume

formula.

Probabilistic interpretations. Supp ose in this paragraph that x ;s 2

S isa probability distribution on S , meaning x 0foralls 2 S and

x =1. Let F denote a random tree distributed according to the forest

volume distribution restricted to forests with a single tree comp onent. That

is to say,foreach tree T lab eled by S , the probabilitythat F equals T is

jT j

P F = T = x

1 s

s2S

where the probabilities sum to 1 over all trees T by 13 for k =1. For

0 p 1 let F denote the random forest obtained by retaining each edge e

of F with probability p, and deleting it with probability1 p, indep endently

as e ranges over the set of jS j1 edges of F . According to [43, Theorem

on F is given by the formula 11 and 43] the distribution of F

jS j1

P F 2 B =p V [F 2 B ] 17

S;1p=p

where

1 k

V [F 2 B ]:=z z V [F has k trees and F 2 B ] 18

S;z S

k =1 7

is a p olynomial in variables z and x ;s 2 S . For for z = x with 0 2= S , this

s 0

p olynomial is x times the volume of trees lab eled by f0g[S whose restric-

tion to S is a forest in B . The probabilistic interpretation of forest volumes

17 makes an imp ortant connection b etween the theory of measure-valued

and partition-valued coalescent pro cesses and the asymptotic structure of

large random trees [6, 4 , 12, 15 ].

Forest volumes of graphs. In the particular case when B is the set of

all forests contained in some directed graph G with vertex set S , Kelmans,

Pak and Postnikov [26 , 27 ] call the p olynomial 18 the forest volume of

G. They obtain a formula for the forest volume of a graph G built up

by a comp osition op eration from simpler graphs, and apply this formula to

obtain some Hurwitz typ e identities by the same metho d of forest volume

decomp ositions used in this pap er. In [26, 27 ] the basic building blo ckfor

explicit formulae is taken to b e the tree volume formula[26,12.1]and[27,

3.2] rather than the forest volume formula. But the forest volume formula

yields Hurwitz typ e sums more easily,ascanbeseenby comparing the

previous discussion around 15 -16 with the treatment of the same result

in [26, x13] and [27 , 4.2]. There seems to b e little overlap b etween the

variations and extensions of Hurwitz's identities found in Section 4 of this

pap er, by consideration of volumes V F 2 B for various subsets B of F ,

S S

and the family of Hurwitz typ e identities found in [27 ] by consideration of

forest volumes of graphs. The basic metho d of forest volume decomp ositions

is doubtless capable of generating still more identities in the same vein.

Other related work. Abramson [2] derived the particular case of Hur-

witz's multinomial formula 16 with x 1;s 2 byKnuth's metho d,

along with more complicated multinomial identities asso ciated with a di-

rected graph. Presumably the results in this pap er could b e also recovered

byKnuth's metho d, but the present approachseemsmuch simpler. See also

Stam [52 ] for another kind of extension of Hurwitz's identities, involving

p olynomials of binomial typ e. Some other pap ers which treat asp ects of

Ab el and Hurwitz identities and their connections to ro oted lab eled trees are

[10, 22 , 49, 50 , 55 , 58 ]. 8

3 Extensions of the forest volume formula

Theorems 2 and 3 in this section presenttwo di erent extensions of the

forest volume formula. Only the rst of these extensions is needed for the

applications to Hurwitz identities in Section 4.

An oriented p ercolation volume. The following formula was discovered

in connection with the problem, solved by Corollary 5 in the next section, of

nding the probabilityoftheevent r ; s for a random forest F distributed

according to the forest volume distribution conditioned on forests of k trees.

Theorem 2 For each R S and each xed choiceofanr 2 R and an

s 2 S R,

jS jjRj1

V [ro otsF =R and r ; s]= x x : 19

S r

Pro of. Let F denote the set of F 2 F with ro ots F = R.For r;r 2 R

S;R S

there is a bijection b etween

F F

0 0

fF 2 F with r ; sg and fF 2 F with r ; sg

S;R S;R

whereby F is derived from F by deleting the rst edge, r ! t say, on the

path from r to s in F , and adding the edge r ! t. Denote the the volume

on the left side of 19 by V r ; s. The bijection gives

S;R

V r ; s=V r ; s:

S;R S;R

Sum this over r 2 R and use the forest volume formula 8 to get 19 . 2

Probabilistic applications. To simplify the next twodisplays 20 and

21 , let

V [ and F has k trees]

P []:=

S;k

V [F has k trees]

where the denominator is given explicitly by 13. Assuming that x 0for

all s, this ratio can b e interpreted as the probability of the event[]for

a random forest F distributed according to the normalized volume measure

on forests of k trees lab eled by S , as considered in [43, 41, 40 ]. Sum 19 9

over appropriate R to obtain for eachchoice of distinct r;s 2 S ,andeach

1 k n := jS j,

x n k

20 P [r 2 ro ots F ;r ; s]=

S;k

n 1 x

and

k 1

P [r 2 ro ots F ;r 6; s]= : 21

S;k

n 1

According to 21, for eachchoice of non-negativeintegers c ;v 2 S with

c = n k , among all forests F of k trees lab eled by S suchthatv has c

v v

v 2S

children in F for every v 2 S , the fraction of F such that b oth r 2 ro ots F

and r 6; s equals k 1=n 1. This rather surprising result can also b e

checked by direct enumeration as in [42, x2].

Adding the right sides of 20 and 21 gives a simple formula for the

probability P [r 2 ro ots F ]. This formula can b e checked by summation

S;k

of the forest volume formula over all R containing r with jRj = k . Obvi-

ous variations of this argument yield formulae for P [ro ots F A]and

S;k

P [ro ots F A]forA S .

S;k

The volume of forests containing a given subforest. Another general-

ization of the forest volume formula with interesting probabilistic applications

[41] can b e formulated as follows:

Theorem 3 For each G 2 F with g treecomponents T G;r 2 ro ots G,

S r

and each R ro ots G,

Y X

g jRj1

jG j

x x : 22 x V [ro ots F = R and F G]=

T G

s r

s2S r 2R

Pro of. This is similar to the condensation argument used in Section 2 to

reduce the forest volume formula to the tree volume formula. For F 2 F

with F G, de ne a G-condensed forest F ,withvertex set ro ots G, by the

following adaptation of Mo on's construction in [34, 37 ]: collapse each tree

comp onent T Gonto its ro ot r , and link these comp onents according to F .

That is, G-condensed F = F where for s; t 2 ro ots G,

F F

0 0

! t for some s 2 T G: ! t , s s

s 10

~ ~

It is easily checked that F 2 F and ro ots F = R. Moreover, for

ro otsG

eachsuch F the volume of F 2 F with ro ots F = R, F G and

G-condensed F = F is easily found to b e

0 1

Y Y

jF j

jG j

@ A

x x

T G

s2S

r 2ro otsG

~ ~

and 22 follows by summation over all F 2 F with ro ots F =R,

ro otsG

using the forest volume formula and

x = x : 23

T G

r 2ro otsG

By summation of 22 over R ro ots G, using 23 , for each G 2 F

with g tree comp onents

g 1

g k

jG j

V F has k trees and F G= x x 24

k 1

s2S

whichwas found by a di erent metho d in [41]. The case of 24 with x 1

is a result of Stanley [53 , Ex. 2.11.a], whichgoesback to Mo on [34] and [37,

Th. 6.1] for k = 1. See also Pemantle [39, Th. 4.2], where a determinantal

formula was found for the probability P T G for T a uniform random

spanning tree of a graph. Other related pap ers are [38 , 28, 14 ]. In contrast

to 24 , there is no simple general formula for V [F has k trees and F G]

for a general directed graph G. But see [26] for expressions of the generating

function 18 of these volumes for graphs with sp ecial structure. Such expres-

sions, which can b e related to 24 by the metho d of inclusion and exclusion,

extend classical results on the enumeration of the numb ers of spanning trees

and spanning forests of G, discussed in Mo on [37, Chapter 6] and Stanley

[53, Ex. 2.11.a].

4 Hurwitz identities

This section presents ve extensions and variations of Hurwitz's identities

2-4 as corollaries of the forest volume formulae provided by Theorems 1 11

and 2. Each identity is derived from its interpretation as a decomp osition of

forest volumes. Throughout the section, is a nite set disjoint from f0; 1g,

and the notation 1 is used. First, an interpretation of Hurwitz's identity

2 as a forest volume decomp osition:

Pro of of 2. Let x = x ;y = x , and multiply b oth sides of 2 by x ,so

0 1 1

that 2 can b e rewritten

jAj1 jAj j j

x x + x x x + x = x x + x + x : 25

0 0 A 1 1 1 0 1

By the tree volume formula, the right side of 25 is the volume of trees

T with ro ot 1 lab eled by f0; 1g[ . The left side classi es these trees T

according to the set A = fv 2 :0; v g of non-ro ot vertices of the fringe

subtree of T ro oted at 0. By applications of the tree volume formula, the

jAj1

factor x x + x is the volume of trees ro oted at 0 lab eled by f0g[ A,

0 0 A

jAj

while the factor x x + x is the volume of trees ro oted at 1 lab eled by

1 1

f0; 1g[A in which 0 is a leaf vertex and the variable x is set equal to 0 to

achieve this constraint. 2

The particular case m = 0 of the next result gives another interpretation

of the same Hurwitz identity 2. See also [40 ] for discussion of further inter-

pretations of 2 in terms of random mappings, including those of Francon

[18, p. 339], and Jaworski [23, Theorem 3], whose equivalence with the inter-

pretations given here can b e explained using Joyal's bijection [24] b etween

marked ro oted trees and mappings [9].

Corollary 4 For 0 m j j

jAj j j

jAj1 jA jm j jm

x x + x x + x = x + x + x 26

0 0 A 1 0 1

m m

= x V [F with m +1 trees :02 ro ots F and 0 ; 1]: 27

f0;1g[

Pro of. The right side of 26 equals 27 by summation of 19 with r =0

over R f0g[ with 0 2 R and jRj = m + 1. So do es the left side, by

decomp osing the volume in 27 according to the set A of all s 2 such that

there is a directed path from 0 to s in F which do es not pass via 1, using

13 . 2 12

As a check, 26 can also b e deduced from its sp ecial case m =0by

replacing x by x + and equating equating co ecients of . The same

1 1

remark applies to the next identity, whose sp ecial case m = 0 is Hurwitz's

identity4.

Corollary 5 For 0 m j j

X Y X

jAj jB j

jAj jB jm jAjm

x +x x +x x +x +x jB j! x =

0 A 1 0 1 s

m m

A B s2B

= x V [F with m +1 trees :0 ; 1]: 28

f0;1g[

Pro of. On the left side F is classi ed by the set A of s 2 such that 0 ; s

and 1 do es not lie on the path from 0 to s,andthevolume is evaluated with

the help of 19 . On the rightside F is classi ed by the set B of s 2 such

that s lies on the path in F which joins 0 to the ro ot of its tree comp onent

in F , and the volume for given B is computed by consideration of the forest

obtained from F by cutting the edges along this path. 2

The last pro of used the idea, adapted from Meir and Mo on [33] and Joyal

[24], of generating a forest by cutting the edges along some path in a tree.

The same idea yields the following identity:

Corollary 6

X Y

jAj1 j j

x + x x + x jAj! x =x + x : 29

A s

A s2A

Pro of. For x = x + x the rightsideisx V [trees F with ro ot 0],

0 1

f0;1g[

by the tree volume formula. On the left side the trees are classi ed by the

set A of all vertices that lie on the path in the tree from 0 to 1. Cutting the

edges along this path makes a forest lab eled by f0; 1g[ whose set of ro ots

is f0; 1g[ A. So the required volume is easily found using the forest volume

formula. 2

As a nal example of Hurwitz typ e sums over subsets, 13 yields easily: 13

Corollary 7 For 1 m j j

jAj1 j j

jAj jAjm j jm

x + x x = x + x 30

0 A 0

m 1 m

which is the the volume of forests F of m +1 rootedtrees labeledbyf0g[ ,

with F classi ed on the left side by the set A of vertices other than 0 in the

treecomponent of F that contains 0.

More exotic Hurwitz typ e identities, involving sums over partitions, can

b e obtained in a similar way.For instance:

Corollary 8 The volume of al l trees labeledbyf0g[ and rootedat 0 is

j j

Y X X

jL j

1 j j1

x = x x + x 31 x

0 0

j 1

j =2

h=1

L ;:::;L

where the inner sum is over al l orderedpartitions of into h non-empty

subsets L ;:::;L , and the trees are classi edaccording to the set L of al l

1 h j

vertices of al l vertices at level j of the treemeaning at distance j from the

root 0 with h representing the maximum height of al l vertices of the tree.

Pro of. This follows easily from the tree volume formula by iteration of the

argument leading to 7. 2

The instance of 31 , with x 1 for all s 2 andx a p ositiveinteger,

s 0

was discovered by Katz [25], who used it to show that the number C n; k

of mappings from an n element set to itself whose digraph is connected with

exactly k cyclic p oints is

nk 1

k 1!kn : 32 C n; k =

As remarked byR enyi [46], the transparentcombinatorial meaning of the

factors in 32 allows either of the formulas 10 and 32 to b e derived

immediately from the other. See Mo on [37 , 3.6] for further discussion, and

[23, 40 ] for the extension of this argumentwhichgives the distribution of the

numb er of cyclic p oints in the digraph of a random mapping s ! M when

the M are indep endent with a common probability distribution which might

not b e uniform. 14

Remarks. The identities in this section were rst obtained in [40] by nd-

ing the probabilityofsomeevent determined by a random forest in two di er-

entways. More identities of this kind can b e gleaned from [40] or derived by

the same metho d. The technique of conditioning on a suitable subtree yields

Hurwitz typ e sums for many other probabilities of interest. A p olynomial

sum of 2 terms is of course hard to compute for large n, but not as hard

as the typical sum of n terms which de nes the probability, and large n

asymptotics of many Hurwitz sums can b e handled as in [5, 4, 8, 12 ].

5 Ab el identities

For x 1 the Hurwitz typ e identities and their interpretations describ ed

in the previous section reduce to corresp onding results for Ab el sums. For

instance, the Ab el typ e identityderived from 30 is

n n k 1 n

k nk m nm

x + k n k = x + n 33

k m 1 m

k =0

for 1 m n. The Ab el typ e identity derived from 29 is the case b =0

of the telescoping sum

nk 1 nb

n x + k x + n =n x + n 0 b n: 34

k b

k =b

Probabilistic interpretation of an Ab el sum. Corollary 5 sp ecializes

for m = 0 to give the following probabilistic interpretation of the Ab el

0;0

x; y de ned by 5, with an asymptotic expression obtained by sum A

a straightforward integral approximation using the lo cal normal approxima-

tion to the binomial distribution [16]:

Corollary 9 For T arandom tree distributedaccording to the forest volume

distribution conditioned on the set of al l trees labeledbyS := f0; 1;:::;n+1g,

with x = x; x = y and x 1 for 1

0 1 s

is a directedpath from 0 to 1 in T is

0;0

x xA x; y

as n !1: 35 P 0 ; 1 =

n+1

n + x + y 2 n 15

In the particular case when x = y = 1, the distribution of T is uniform on

n+1

the n +2 ro oted trees lab eled by S , so 35 implies that for all distinct

u; v 2 S := f0; 1;:::;n +1g

0;0 n+1

P u ; v =A 1; 1=n +2 : 36

0;0

So A 1; 1 is the numb er of ro oted trees lab eled by S in which there is a

directed path from u to v . It is easy to deduce from 36 the result of Mo on

[35, Theorem 1], that the conditional exp ectation of the size jfv :0 ; v gj of

the fringe subtree of T with ro ot 0, given that T has some ro ot other than 0,

0;0 n

. The asymptotic evaluation in 35 agrees with Mo on's is A 1; 1=n +2

asymptotic formula n=2 for this conditional exp ectation. As observed by

Mo on, this conditional exp ectation is also the exp ected distance in T between

anytwo distinct vertices u; v 2 S . See [8] for a study of the asymptotic

b ehaviour for large n of the distribution of the size of the fringe tree fv :

0 ; v g for T distributed according to a non-uniform volume distribution on

trees lab eled by S , and [4, 12 ] for further study of the asymptotics of large

random trees of this kind.

Acknowledgment

Thanks to Jurgen Bennies, Ira Gessel, Richard Stanley,Volker Strehl

and some anonymous referees for helpful suggestions and remarks on earlier

versions of this pap er.

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