DOCUMENT RESUME

ED 472 095 SE 066 382

AUTHOR Labelle, Gilbert

TITLE Manipulating Combinatorial Structures.

PUB DATE 2000-00-00

NOTE 18p.; In: Canadian Mathematics Education Study Group = GroupeCanadien d'Etude en Didactique des Mathematiques. Proceedingsof the Annual Meeting (24th, Montreal, Quebec, Canada, May26-30, 2000); see SE 066 379.

PUB TYPE Guides Non-Classroom (055) -- Speeches/Meeting Papers (150)

EDRS PRICE EDRS Price MF01/PC01 Plus Postage.

DESCRIPTORS Curriculum Development; Higher Education; InterdisciplinaryApproach; Mathematical Applications; *Mathematical Formulas;Mathematical Models; *Mathematics Instruction; ModernMathematics; Teaching Methods

IDENTIFIERS Combinatorics

ABSTRACT

This set of transparencies shows how the manipulation ofcombinatorial structures in the context of modern combinatorics can easilylead to interesting teaching and learning activities at every level ofeducation from elementary school to university. The transparencies describe:(1) the importance and relations of combinatorics to science and socialactivities; (2) how analytic/algebraic combinatorics is similar toanalytic/algebraic geometry; (3) basic combinatorial structures withterminology; (4) species of structures; (5) power series associated with anyspecies of structures; (6) similarity of operations between power series andthe corresponding species of structures; (7) examples illustrating where thestructures are manipulated; and (8) general teaching/learning activities.(KHR)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

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Manipulating Combinatorial Structures

Gilbert LabelleUniversite du Quebec a Montreal (UQAM)

A combinatorial structure is a finite construction made on a finite set of elements. In real-lifesituations, combinatorial structures often arise as "skeletons" or "schematic descriptions"of concrete objects. For example, on a road map, the elements can be cities and the finiteconstruction can be the various roads joining these cities. Similarly, a diamond can be con-sidered as a combinatorial structure: the plane facets of the diamond are connected togetheraccording to certain rules.

Combinatorics can be defined as the mathematical analysis, classification and enu-meration of combinatorial structures. The main purpose of my presentation is to show howthe manipulation of combinatorial structures, in the context of modern combinatorics, caneasily lead to interesting teaching/learning activities at every level of education: from el-ementary school to university.

The following pages of these proceedings contain a grayscale version of my (color)transparencies (two transparencies on each page). I decided to publish directly my trans-parencies (instead of a standard typed text) for two main reasons: my talk contained a greatamount of figures, which had to be reproduced anyway, and the short sentences in thetransparencies are easy to read and put emphasis on the main points I wanted to stress.

The first two transparencies are preliminary ones and schematically describe: a) theimportance and relations of combinatorics with science/ social activities, b) how analytic/algebraic combinatorics is similar to analytic/ algebraic geometry. The next 7 transparencies(numbered 1 to 7) contain drawings showing basic combinatorial structures together withsome terminology. Collecting together similar combinatorial structures give rise to the con-cept of species of structures (transparencies 8 to 12). A power series is then associated to anyspecies of structures enabling one to count its structures (transparencies 13 to 15). Each op-eration on power series (sum, product, substitution, derivation) is reflected by similar opera-tions on the corresponding species of structures (transparencies 16 to 18). The power of thiscorrespondence is illustrated on explicit examples (transparencies 18 to 26) where the struc-tures are manipulated (and counted) using various combinatorial operations. Transparen-cies 27 to 30 suggest some general teaching/ learning activities (from elementary to advancedlevels) that may arise from these ideas. I hope that the readers will agree that manipulatingcombinatorial structures constitutes a good activity to develop the mathematical mind.

References

The main references for the theory of combinatorial species are:

Joyal, A. (1981). Une theorie combinatoire des series formelles, Advances in Mathematics, 42, 1-82.Bergeron, F., Labelle, G., & Leroux, P. (1998). Combinatorial Species and Tree-like Structures.

Encyclopedia of Math. and its Applications, Vol. 67, Cambridge University Press.

BEST COPY AVAILABLE

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e is

of a

F-s

truc

ture

and

aci

lsjo

irrt 6

-str

uctu

re

The

spe

cies

F(G

) =

Fo

Ga

F.-

stru

ctur

eA

(F

o G

)-st

ruct

ure

isbu

ilt o

n a

finite

set

of

disj

oint

G-s

fruc

ture

sF

OG

(her

e, G

( 0-

0)

The

spe

cies

F'

A F

'- st

ruct

ure

on U

isra

F-s

truc

ture

on U

+ f4

ti

TH

EO

RE

M. I

f F a

nd G

are

spe

cies

, the

n(F

+G

)(x)

= F

(x)

+ G

(z),

(F-

= F

0c)

GO

O,

CF

o 6)

(21=

F(G

Oe)

), C

ifF

'Cx)

=F

rac)

.

EX

AM

PLE

S /

AP

PLI

CA

TIO

NS

1) E

= E

.* E

odd

E (

X)

- E

even

CX

) E

oda

OC

)

that

is,

ez=

cos

t,x

sinb

2) L

et D

be

the

spec

ies

of d

eran

gem

ents

,i.e

., pe

rmut

atio

ns w

ithou

t fix

ed p

oint

s, th

enS

E D

mss Afr

ls.

S(x

)=-

EC

x)D

(x)

-riz

a- =

ex

3) C

x.)

Dw

r.:

d =

num

ber

of d

eran

gem

ents

of

n ob

Ject

s

Ot 7

21

31

A.

1920

3)E

nd =

S °

A

4?lc

"*

4e.

)41

(ol

441

r414

'.0

4.-

:(4*

)4

?/.

C07

./._o

va;

/ --C

ick-

,04-

`.4t

.4,

®17

--0 /

\E

nci(x

)-z.

S(k(

x))

E" n

-rn

s.- 1-

Ak)

n"N

alP

I!

AM

=yi

nz"

Tv

n

LQ

....."

..,...

.

,4

,f)

\,),

k..)

,,om

u.t-

o),.;

*a4,

/.

f4.

-

s(x)

= E

(c(1

))ix

Eva

)41

60=

(m-

)!

4b)

Let

SO

> b

e th

e sp

ecie

s of

per

mut

atio

nsha

ving

* c

ycle

san

dE

kbe

the

spec

ies

of *

-set

s,

then

S s

Eko

44,

Edt

(x)=

S(*`

)(x

)"1

6 E

it(03

e1)

rg,,

A,.

=nu

mbe

r of

per

mut

atio

ns o

f n p

oint

s ha

ving

k c

ycle

s

abso

lute

val

ue o

f Stir

ling

num

ber

of th

e fir

stki

nd

=E

no,,,

..tle

f+

VII

> 0

,41(

rt,1

01

r

<7,

2122

5a)

33=

--E

0E+

Whe

re E

. is

the

spec

ies

ofno

n-em

pty

sets

.

E+

(x)

=e

i =B

C3e

)= E

(E+

Cx)

)

=ee

e''

ee

It fo

llow

s th

at6=

num

ber

ofpa

rtiti

ons

of a

set

of n

ele

men

ts

I

E.

L.n

Thi

s is

Dob

insk

i's fo

rmul

ae

.fo

r th

e n

th B

ell n

umbe

r4m

0

5b)

Let .

B<le

>be

the

spec

ies

of p

artit

ions

havi

ng *

cla

sses

and

Eli

be th

e sp

ecie

s of

*-s

ets,

<4e

>B

= E

lio(2

3= [

num

ber

of p

artit

ions

of n

poi

nts

into

A c

lass

esS

tirlin

g nu

mbe

r of

the

seco

nd k

ind

Igr

i*I

riin

nr)

'rh

t*

YIL

> 0

6)=

E. E O

N 2

= (

E4

ex

num

ber

of s

ubse

ts o

f a s

et o

f r, e

lem

ents

2"(k

now

n!)

A7)

Let S

be

the

spec

ies

ofpe

rmut

atio

ns^

havi

ng o

nly

even

cyc

les

:S

.--

E0C

even

m.e

c.,..

(2).

. eil.

,Fc

,

V7-

77

z, .

is 3

2. s

t..af

t-it

if h

= 2

k.,

0ot

herw

ise

8)-

efil

140.

414

est

L'(x

)=(L

fr))

t=

xl(k

now

n!)

2324

9)C

;=L

(aga

inI

)

lid(X

)1=

144

4.00

dx

Lod

d=

LI

even

but

Pb'c

even

12)

A=

X' (

E°A

)

I./

= -

= X

(xl-

E00

0)z e

ik(x

)A

(30

4.14

x)X

ann

11)

B1

E B

recu

rren

ce)

IliB

(2)

1=E

(L)B

0011

, ex-

33(z

)bh

. I(*

)*m

e

13)

Let

be th

e sp

ecie

s of

ver

tebr

ates

( a

vert

ebra

te is

a p

oint

ed r

oote

d tr

ee )

= L

OA

,.,i,

.,0f

/...

......

...._

__\.a

. a c

e....

.,....

:fr

(ze

esi

," 0

1_,A

/_.:

e--.

. .4

\se

s-..

4....

..-w

,...

..., /

4,

.-._

....

as(5

-o-*

il,-,

.....

...

I. v

e(x)

.. L

(A

(20)

5 -

1_A

(.)

5E

A44

4Cx)

A/4

..r.

n a

. r.

num

ber

of e

nclo

func

tions

on

P e

lem

ents

n"a.

r. n

" -I

(Joy

al)

7:V

2526

14)

gl=

the

spec

ies

of fo

rest

s of

roo

ted

tree

s

a'=

41'

..

1/4.

fe-7

,1.

4k.

Hen

ce, t

he n

umbe

r of

bal

lot o

utco

mes

for

nca

ndid

ates

Is

*"

16)

Let 8

4 be

the

spec

ies

of b

allo

ts.

43.0

1 =

L0

Ell.

&LI

N L

tE.6

0 =

L (

ex-

i)as

430

(7);

'4-

----

--T

er5u

-kt5

+6.

1)11

760

claw

nil

titQ

,nu

mbe

r of

fore

sts

of r

oote

d tr

ees

onn

poin

ts,o

v-I

=+

I j

15)

Aul

t=

Et

21,1

4(z)

,..e

it'`,

''`)

Zia

(29

=L

t A. (

x.)

= L

te,)

,T.;

inintegerY)

17)

Inv

= E

a+

E,,)

"1.1

1

oi0*

* ef

f

IL.

"0

Ilith

eiV

oirn

ifbite

e:vv

r+0:

34x1

n:ol

eutl%

en4:

onn

elem

ents

Is

al)1

.3.5

-42*

-0

18)

Cla

ssic

al fo

rmul

as o

f Cal

culu

s ho

ld

For

exam

ple

: (F-

0 6,

Y=

(Fio

&)G

' [ch

ain

rule

]

G V

G -

Rgb

rG

)(v-

"N

.,

11cs

;LN

.,0

-*(r

*aF.

6G

.R::

oJC

's

2728

PO

SS

IBLE

TE

AC

HIN

G /

LEA

RN

ING

AC

TIV

ITIE

S(

from

ele

men

tary

to a

dvan

ced

)

To

find,

bui

ld o

r dr

aw c

ombi

nato

rial

stru

ctur

es fr

om r

eal-l

ife s

ituat

ions

or fr

om c

oncr

ete

/ abs

trac

ts o

bjec

ts.

To

deci

de w

heth

er tw

o st

ruct

ures

of a

giv

en s

peci

es a

re e

qual

or

not.

To

mak

e a

com

plet

e an

d no

n-re

petit

ive

list o

f str

uctu

res

belo

ngin

g to

a g

iven

spec

ies,

labe

lled

by a

giv

en (

smal

l) se

t.

To

mak

e a

com

plet

e an

d no

n-re

petit

ive

list o

f unl

abel

led

stru

ctur

es b

elon

ging

to a

giv

en s

peci

es o

n n

indi

stin

guis

habl

eel

emen

ts (

n sm

all).

To

find

com

bina

toria

l equ

atio

ns a

ssoc

iate

dto

spe

cies

(an

d vi

ce-v

ersa

).

To

find

enum

erat

ive

or s

truc

tura

l pro

pert

ies

of s

peci

es fr

om th

eir

com

bina

toria

l equ

atio

ns(v

ia s

erie

s ex

pans

ions

, for

exa

mpl

e).

To

find,

bui

ld o

r dr

aw c

ombi

nato

rial

stru

ctur

es fr

om r

eal-l

ife s

ituat

ions

or fr

om c

oncr

ete

/ abs

trac

t obj

ects

.

Riv

ers

mak

ing

a (p

lane

) tr

ee

Flo

wer

s

1014

Vis

:47

Sitt

ing

arou

nd a

rou

nd ta

ble

2930

To

deci

de w

heth

er tw

o st

ruct

ures

of a

giv

en s

peci

es a

re e

qual

or

not.

s-b-c-d-e

f-g-h-I-J

a-g-h-l-J

I1

1i

f-g-h-i-J = a-b-e-d-o * f-b-c-d-e,

To

mak

e a

com

plet

e an

d no

n-re

petit

ive

list o

f str

uctu

res

belo

ngin

g to

a g

iven

spec

ies,

labe

lled

by a

giv

en (

smal

l) se

t.

The

16

tree

s la

belle

d by

{ a

, b, c

, d }

:a-

b-c-

da-

c-b-

da-

b-d-

ca-

d-b-

ca-

c-d-

ba-

d-c-

b

b-a-

d-c

b-d-

a-c

b-a-

c-d

0-c-

a-d

c-a-

b-d

c-b-

a-d

b\a/

dI G

b\c/

a

I d

cd

b I ab\d/ a

I G

To

mak

e a

com

plet

e an

d no

n-re

petit

ive

list o

f unl

abel

led

stru

ctur

es b

elon

ging

to a

giv

en s

peci

es o

n n

indi

stin

guis

habl

eel

emen

ts (

n sm

all),

The

2 u

nlab

elle

d tr

ees

on 4

iden

tical

ele

men

ts:

(0-4

,-0-

411

Y

To

find

com

bina

toria

l equ

atio

ns a

ssoc

iate

dto

spe

cies

(an

d vi

ce-v

ersa

).

To

find

enum

erat

ive

or s

truc

tura

l pro

pert

ies

of s

peci

es fr

om th

eir

com

bina

toria

l equ

atio

ns(v

ia s

erie

s ex

pans

ion,

for

exam

ple)

.

<St-- OW, 7q

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