Labeled trees and relations on generating functions - Springer Link

LABELED TREES AND R E L A T I O N S ON GENERATING FUNCTIONS M.P.

DELEST,

J.M.

FEDOU

Bordeaux

I University LABRI ÷ of Computer Science

Laboratory

Abstract.

In interpretation by R. Stanley.

this paper, for a property Our proof is

kind

trees and forests. donnons ici une

of

labeled Nous

R~sum6.

d'une propri~t~ Stanley. Notre d'arbres

et

de

we give on g e n e r a t i n g based upon the

des fonctions preuve utilise forSts

a

combinatorial functions gived study of special

interpretation

combinatoire

g~n6ratrices une classe

circe par particuliere

R.

~tiquett#es.

INTRODUCTION Trees are presents in There are the natural

science.

such as expressions

many subjects representation

programs, arithmetic in language theory.

The

structure

of

complexity

of

structure precisely

which allows the complexity

[4][6][11]). the paper

algorithm

Trees Viennot

of

theorical of a lot

expressions,

tree

is

because to

of

very

they

words

efficient

in

constitute

a

organize informations of algorithm (see

are also [14].

present

in

other

of

computer objects

or

algebraic

the

study

dynamic and to for

fields

as

of data

measure example shown

in

In some purely combinatorics subjects, they are the nice objects for understanding formulas. For example labeled trees, as defined in [2], are usefull in many subjects, see for instance Cori and Vauquelin [3] for the construction of a bijection between planar graphs and well labeled trees, or Moon [9] for identities

on

basis of and trees

forests.

Moreover,

In this paper, we introduce trees, the k-shaped forests, in

labeled proof

labelled

and

a

÷ Unit6 de Scientifique Post FRANCE.

the

new theories in combinatorics (see are very studied in this context.

generalization

Recherche n°~26.

Mail

Electronic This work Informatique'.

:

Associ6e

351Cours

Mail was

of

de

a

result

au la

Centre

bijections

for

are

example

the

[7],[13])

a special kind of set of order to give a bijective about

generating

National

Liberation,

de

33405

functions

la

Recherche

TALENCE C e d e x ,

: maylis~geocub.greco-prog.fr supported

by

the

"PRC

de

Math~matiques

et

194

given by R. Stanley this paper). In fact,

we

in [10] (see proposition 19 a t study the generating functions

the

end

F~(X) = ~ Nk(n) X n, where ~)o Nk(n ) = ~(f~+fn+...+fn )(fn+f.+...+fn )...(fn+fn+...+fn 1

R.

Stanley

k=2,

saying

2

k

gives that

the the

2

k÷l

3

result

formula

for for

k=2

k=3

S

and

3,

appears

$+|

of

). s+k-1

the

proof

after

an

only

for

enormous

amount of cancelation and tedious computation. Moreover he says that he does not known a simpler alternative method. We give it in paragraph 3 a n d i n s o m e w a y we g e n e r a l i z e his result. After some definitions and notations, in section 2 and 3, we b r i n g back the problem of the determination of F~(X) to the enumeration of k-shaped forests according to the number of sons of each vertex. in the system

This problem of enumeration is then particular cases where k=2, 3 and 4. of equations obtained by the mean of

The readers methods.

1.

are

DEFINITIONS

refered

to

[4][5][14]

solved, in section 4, We solve a linear the k-shaped forests.

for

examples

of

similar

AND NOTATIONS

tree is a connected graph G without cycle [1] vertex r called root. If (f,s) belongs t o G, f (resp. s) is called father (resp. son) of s (resp. f). Here, we will consider trees which have sometimes a loop on the root. A set of trees is called a forest A /abe/ed tree is a tree with an integer, called label, associated to each vertex. The entering degree deg(i) of a vertex labelled i is the number of sons of this vertex. The difference degree ~(i) of a vertex labeled i is the difference between the label of the father of this vertex and i. The difference degree of a loop is equal to 0. It is easy to generalize these definitions to the forests. We n o t e ~ the empty forest. with

A rooted a distinguished

Let K be an half ring and Y be {Yl, Y2, ''', denote by K[Y] the ring of the formal power series over coefficients in K. Let f(x)=~,~ofnx" and g(x)=~,~og,x" formal power series from K[{x}], the product of Hadamard g is defined by f~g(x) Generaly Hadamard

p

convention, We deg

call and

speaking, times of we n o t e A =

we the

= ~o

will serie

(~f)o

fng. x""

denote f that

= ~>Ix

parameter an application 6 are parameters.

y k } . We Y with be two of f and

by (~f)P is (~f)P =

the ~n~c

product f, pxR"

of By

~ = x/(l-X). from

~

into

~.

The

two

degrees

195

Q

/

/

3

1

7

6

/\ 4

/

5

2

Figure 2.

K-SHAPED We

introduce

to

"explain"

products

Definition to

t.

1.

A forest

~3,8"

of

labeled

FORESTS a

special

such

as

kind

forest,

in

order

(yl+...+yk)...(y~+...÷y,,~_l).

1. Let ~ be a forest forest ~ is k-~abe~ed 0 ~ ~(i)

The

of

having t when, for < k .

vertices euery i

[abe~ed in [1..t]

from

1

Definition 2. A k-Cabe~ed forest ~ is said k-shaped if ~ has s+k-1 vertices with s~O s u c h t h a t i) every root in ~ ~abeLed with 1 in [1..s] has a Coop, ii) the k-1 ~ast vertices ~abe~ed from s+l to s+k-1 are roots without Coop, Notations. ~.~

is

the

set

of

k-shaped

forests

having

s+k-1

vertices. ~k.0 is the unique forest made we d e n o t e it by e. ~K i s t h e s e t o f k - s h a p e d forests.

vertices,

See

for

example

Remark mean of

the

figure

S. Let ~ be the following

in

12

the

~k,s, we algorithm:

forest

obtain

~ k

with

k-1

belongs

forests

to in

isolated

~3,e*

~k,s,l

by

the

to

k.

is

k".

begin for

every

Let

1 be (i)

vertex the

in

new

either

a

(LL)either

~

do

vertex; loop

a son

label then

(vertex):=label 1

(vertex)+1;

is

, of

one

of

the

vertices

numbered

2

end

Thus,

the

number

of

k-shaped

forests

having

n+k-1

vertices

196

2

;

2

/

3

;

1

/

3

;

/ /\ 1

1

©1

/ /\

5

6

2

/ /\

4

2

2

©

o

;

;

7

6

/

3

3

/ / /\ /

1

5

3

4

4

The figure using this Remark

4.

2.

An e x a m p l e

P be

8

;

5

2

of

the

algorithm

2 shows the construction of algorithm. In this example, Let

7

6

1 Figure

;

3

2

/

5

4

the

of

remark

the forest we h a v e k = 3 .

3

of

figure

1

product

(YI+Y=+Y3)(Y2+Y~+Y4)(Y~+Y4+Y~)(Y4+Ys+Y6)(Y~+Y6+Y~)(Y6+Y~+Ys). In

some

way,

the

term

y32y4y~y~

we

can

say in

that

the

forest

every

relation

the

i th

factor

(yl+yi+z÷yi+2)".

Let

p be

the

forest

is

father

5.

The

monomia~

weight

defined

figure

P.

of

i"

to

p of

In

means

1 "represents" fact, "we

on

choose

this yj

in

according

For example, according to

if the

a

forest

~

from

~,~

by

gp(~)(Yl,''''Y~+k-1)

are

of of

a parameter.

Definition

£s

"j

the

development

A is the forest parameters entering

'11

=

of degree

figure lj the and difference

repeetively ~de~(~)=Y32Y4Y6~Y~,

and E6(~)=y12y2~y4~y~ys

Yi

.

weights degree

197

Notation. into ~[Y]

For every defined by

forest

~

from

Yk,~,

let

~ be

the

map

from

~

s+k-1

i=l Define

the

formal

power

serie

¢(Yk)

T ¢ ( R k) i s t h e g e n e r a t i n g to the number of sons

~(Yk)

of

of

by

~(~).

function of the vertices

=

2[Y]

ni

the k-shaped that is

...i 0

forest

Yo°''-Yk

k

~ccordiag

•

k

ioj...~ik_ I where

n I

...~ 0

for

each

j

is

the

number

of

k-shaped

forests

such

that,

k

in

[O..k],

there

is

i

vertices

which

have

j

sons.

J

Definition

6.

and ~ be an into 2 [ x ] d e f i n e d

K[x]

f(x)

Let

e~ement by

of

At(~)

P r o p e r t y 7.

A t and

~eee

= ~n>l ~,s"

f . x ~ be a forma~ power serie We d e n o t e by A r the map from

=~T (.f)a~g(i) l~i~s+k-1 are

morphisms

We obtain At(~) substituting value (zf) in Hd~z(~) or the value (.f) (u,~'} be a forest where u is a tree the previous definitions to the labeled equalities

which

prove

of

~k

the

property

Af({u,~'))

over

i

the

forests.

formally t o e a c h Yi t h e in ~(~). Moreover, let a n d ~" a f o r e s t , applying trees gives the following

7

-- A f ( u )

Ar(~')

and 11~((u,~}) Examples: In ~ k , I, w e h a v e A f ( ~ )

= IIde~(u)

IId.~(~' ).

we h a v e A f ( ~ ) = A k-1 If ~ is the f o r e s t = A 4 f ~ ( z f ) 2 a n d ~(~) = Y o 4 Y z 2 Y 2 2

of

figure

198

8. are

Definition

~

and ~

L e t ~1 and ~2 be two f o r e s t s equivatents when, ~f n

there

exists

a permutation

The

forests

n

s÷k-i

1

then

from Yk,~"

a of

such

[1..s+k-1]

that

n

o(1)

The

property

Proposition We h a v e

the (i) (ii)

7

gives

~k

that

OF

this

Fk

IN

two e q u i y a l e n t

10. The function ~(~k) of Yi by ( * f ) ~ we give k,s and

k-SHAPED

we g i v e

some

precision

about

R

Fk(X)

using

the enumerating forma/y each

Define,

for

every

2

3

:

),

)...(f.+f.+...+f. S

k+1

ordered

all

S+I

S+k-I

partitions

of

~n~IN~,s(n) x"

we h a v e =

~s~O Nk, sCn), Fk(x>

=

~s~o

Fk,

s'

also

Fk,s(X)

= ~ Pk

(fr

"

'fp

1

where

to

F k,

)(f,+f,+...+f,

2

Nk(n) and

for

n

F~,s(x) that

from ~x,s"

FORESTS

an expression

where the sum is taken over n:nl+n~+...ns+~_ 1 w i t h n i ~ l and

Note

forests

function Fk(x) is deduced from the k-shaped forests substituting

ECfn+f.+...+f. 1

T E R M OF

paragraph,

Theorem

N~.s(n)=

proposition.

following

~(~) = ~(~2), if f E K[x], At(~l)=At(~2).

INTERPRETATION

First, integers

the

9. L e t ~1 and ~ be fotLow~ng properties

3.

In is

immediatly

a(s+k-l)

1 if

the

sum

is

over

all

Let Pk,s(Yl,''',Ys,k-1) s=O a n d o t h e r w i s e given

Pk, s ~ Y I , . .

o,Ys+k_~)

ordered be by

)x~ s÷k-I

the

partitions

of

polynomial

= ( Y l " I ' . - oYk) ( Y 2 + . o . + Y k + x ) . . .

of

n with Z [Y]

ni~l. equal

(Ys+...+Ys+k-1~

•

199 In

order

to

prove

the

theorem

10 we p r o v e

the

following

V--"

Lemma 11, P~,s ( Y l ' ' ' ' Y s + k - 1 )

=

~__

K d e g ( ~ ) ( Y I ' Y s ' .... Y s + ~ - 1 ) '

~E~k,s If s=O, that

the

result

is i m m e d i a t e . k

: ~_

Pk,~(YI,''',Y~+k-I)

Otherwise,

an

induction

shows

YiP~,s-~(Ys,''',Y~÷k-1)

i=l k

¢ )__

~_

YtKdeg ( ~ ' ) ( Y 2 ' ' ' ' ' Y s + k - 1 ) '

Using the algorithm of construction ~k,5 c o m e s f r o m a forest ~" of ~ k , s - 1 a loop in ~,

of ~k,sJ each forest If the v e r t e x l a b e l e d

Hdea(~)(Yl,...,Ys+k-1)=YlHde~(~') Else,

the

vertex

labeled

1 is

a son

of

(Y2,...,Ys+k_I). a vertex

i

Hde~(~)(Yl,...,Ys÷k_l)=YiH~eg(~')(Y~,...,Ys,k_l)Thus

the

lemma 11

is

proved.

We d e d u c e

I

n~l and

thus

by

p1+..+ps+~_1=n

factorisation,

s÷k-1

~E~P~ . ~

we o b t a i n s+k-1

i=l

El

deg(i) p i

Pill

Consequently

we h a v e Fk, ~ (x)

=

A f (~).

,~E~'~,

~ of 1 is

X

1 i

in

[2..kS

and

200

Using

the

notation

F~(x)

= ~ Fk,s(x),

we

get

s)O

F~(x)=~

Af(~)

~E~ k and

4.

theorem

10

follows

from

property

RELATIONS BETWEEN SUBSETS OF ~ In

subsets

this

which

power serie are given by

section,

we

basic

properties

@(~k)

as

sum of

=

formal

we

provide

obtain -

two

lemmas with

relations

types

12

to

respect

of

in

16

15

to

Most deduced

allows apply

of

the

Notations:

series

Pi

i

formal

...i

whose

~, Ei

we

us

us

power

series

to

express

and

P ......... I

to

write 12

to

lemmas 8)

are

of

proposition

positive by

...i

p

p+q

P~

...) 1

upon

types

of

the

p÷q relation

forests

and

of are

9.

integer

of

as P+q

based

particular

every

set

Pi 15. 1

of

denote the

formal

J

lemmas

these

"''i

these

p+q

application

For

[O..k-1]

allow

...i

(definition

by

power

particular the

p

between

will

Pi

to

will

order

equivalence

into

relations:

p

lemma

~k

decompose

...i I

Then,

set

to

~(~)

) L__

£

~EEi

we

the us

1 2

i .. .i 1

part allow

\ Pi

in

7.

p,

forests

and

belonging

for

every

to ~ k

p-uple

having

at

1

least

"

ei

...i

p+k-1

vertices

the

forest

such of

~x

that

V

j E

with

p+k-1

[1.,p], vertices

~(j)=ij, such

that

1

V j E [1..p], By

conventions

Example: belongs

to

the the

let

E

(resp.

e)

6(j)=ij.

be ~

forest ~ of the sets EjE2,E~2jE220,

(resp.

example ....

1

~,o)" is

exactly

e22o211

and

201

For euery

Lemma 12.

p-up~e

(il,i~,..,i

of [ O . . k - l ] p,

p) k-1

Ei

1 ...i p

=

ei i . . . i p

+

~

Ei i ...i p h

•

(I)

Pi 1 ...i

p h-

(2)

h=O

and

k-i

Pt I ...i

p=

~o ( e i i . . . i

P)

+

~ h=O

Indeed~

a

forest

of

Ei

...i

is

1

-

either

the

p

forest

e~

.

.

.

i

1

or

-

a

half

forest

having

at

degree

of

vertex

the

:P

p

least

p+k p+l

vertices~

is

in

k-i

Examples:

such

that

the

[O..k-1]. k-I

E=e+

E i

,

@(~)

=

Yo

+

Pi,

i=O

i=O

k-I

k-I

Ei=ei+

Eij

'

Pi

=

Yl

Yo

÷

Pij

j=O

j=O P

Lemma

13.

Pp(p-1)...zo

Proof:

An

{uj~'}

where

loop

and

p

element u

is

sons

= Yo Y p , z

~ the

of

E~(~_l)...1

labeled

tree

from

1 to

labeled

C~(~k)

~k where each label 1 of a vertex T h u s we h a v e t h e a n n o u n c e d result

(4)

(see

o whose

p and

figure

root where

has been using the

3)

labeled ~"

is

any

changed property

in 7.

p+l

1

p U

Figure

3.

An e l e m e n t

of

is

E~(p_l)...1

o.

a

p+l

forest has

forest (t+p)

a of

202

In

the

same

Lemma 16. Example: and also Lemma

way,

we

fix

j

P~