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)
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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
2 3
PR
ELI
MIN
AR
IES
The
dyn
amic
con
nect
ions
bet
wee
n
com
bina
toria
l / d
iscr
ete
scie
nce
I soc
ial
mat
hem
atic
san
dac
tiviti
es
will
gre
atly
incr
ease
dur
ing
the
com
ing
deca
des
:
Com
pute
r sc
ienc
e(a
naly
sis
of a
lgor
ithm
s. o
rgan
izat
ion
and
com
pres
sion
of d
ata
stru
ctur
es, e
tc)
Che
mis
try
(cla
ssifi
catio
n. s
ymm
etrie
s of
mol
ecul
es,
isom
ers.
etc
)
Com
bina
toric
s
Phy
sics
(cla
ssifi
catio
n, s
ymm
etrie
s of
ato
ms,
quan
tum
theo
ry. F
eynm
an in
tegr
als.
etc
)
Bio
logy
(ana
lysi
s of
DN
A, g
enet
ic c
ode.
clas
sific
atio
n of
spe
cies
, etc
)
Mat
hem
atic
s(a
naly
sis,
cal
culu
s, a
lgeb
ra, g
eom
etry
,gr
aphs
, gro
ups,
etc
)
Hum
an a
ctiv
ities
(pub
lic k
eys
met
hods
of e
ncry
ptio
n,en
codi
ng o
f com
pact
dis
cs.
secu
re tr
ansa
ctio
ns, c
omm
unic
atio
ns.
desi
gn o
f pla
ne fl
ight
s, s
hort
est p
aths
.te
achi
ng a
nd le
arni
ngba
sic
com
bina
toria
l stu
ctur
es. e
tc)
AN
ALY
TIC
/ A
LGE
BR
AIC
GE
OM
ET
RY
Geo
met
rical
-4M
athe
mat
ical
figur
esfo
rmul
as
02x
- e
y +
7 =
0
x2 +y2
= 9
y=e2
AN
ALY
TIC
/ A
LGE
BR
AIC
CO
MB
INA
TO
RIC
S
Com
bina
toria
l--
->M
athe
mat
ical
stru
ctur
es4-
-fo
rmul
as
0Y
=1"
*()
y =
x +
xy2
e3x
12
MA
NIP
ULA
TIN
G C
OM
BIN
AT
OR
IAL
ST
RU
CT
UR
ES
Gilb
ert L
abel
le /
LaC
IM -
UQ
AM
In C
ombi
nato
rics,
a s
truc
ture
, s, i
sa
finite
con
stru
ctio
nm
ade
on a
fini
te s
et U
.
We
say
that
: U
Is th
e un
derly
ing
set o
fs
or: U
is e
quip
ped
with
(th
e st
ruct
ure)
sor
: s is
(bu
ilt)
on U
or: s
is la
belle
d by
the
set U
Exa
mpl
e 1.
A tr
ee o
n tJ
=a,
b, c
,,
+
el"
kit
"1:\6
cif
JR
ay
Exa
mpl
e 2. 4
A r
oote
d tr
ee o
n 1.
7..k
bc,..
x,rt
+3
e
1,4
3 a
a.
4
Exa
mpl
e 3.
An
orie
nted
cyc
le o
n II
f,
936
IA
SP
'kcL
-.1
(4...
;.7
3V
aria
nts:
even
(or
ient
ed)
cycl
eod
d (o
rien
ted)
cyc
le
Exa
mpl
e 4.
A s
ubse
t on
(of)
tro
1%.f
f.,
(or
bico
lora
tion)
0_L
0
34
Exa
mpl
e 5.
A (
the)
set
str
uctu
re o
nE
xam
ple
8.A
par
titio
n on
= 1
0,*,
42,
3, 4
, (7,
'1'3
1:1
U /,
0AA
J- TJJ1 Var
iant
s:ev
en s
etod
d se
t
Exa
mpl
e 6.
A p
erm
utat
ion
onLY
-=Ii
2/30
0 /6
, ai b
d
ys..,
.ITt
.egk
tN,
.".`
tf
ts
. \--i
i.el
;0U
',.,
..*
.11
.-1N
e4
r )%/
' I;
/\-
- 4.
a6
i ---
>...
. f
Exa
mpl
e 7.
Alin
ear
orde
r on
U=
1=d
Al;
.40;
a3
er xs44
.
Exa
mpl
e 9.
An
endo
func
tion
on62
-3g,
aS
e34
V24
1g1 1-
3 e
2}ri
Vr.
.v,.
ea9
NN a
S.
H.
Of
fjR
io63
1S
y
r17.
-43"
4
Iffa
r.v.
../6
2(iv
itvy
'IVts
,62,
44,
Soc
/4 I
ti,.4
7I
17
0
2.,,
,..\,.
..0.,.
......
..0...
4".
Ft 3
s:-"
,3..,
..-er
-'-er
e...
a 3
2N
5?
SP
*3
56
Exa
mpl
e 10
. A b
icol
ored
pol
ygon
on
TJ=
14,1
2,c,
d,e,
f3
Exa
mpl
e 11
.A
gra
ph o
n E
T=
it,
.. .
,fq
,,-r,
;--,
-:--
-7
0Z
S
/3,2
4,...
......
.___
#07
.t.,
11
N,
ir...
.,,,,,
,.....
..4T
4....
.1 -
..A
teip
23
9...
.Ie.
'11
-:1V
,,..
.....,
.---
2 9
r, fvS
-2
a.,
q5'
i..g
Exa
mpl
e 12
. A c
onne
cted
gra
ph o
n U
=
ft'
i9n
"111
'
,00.
417
34/
6
Exa
mpl
e 13
. A s
ingl
eton
on
U=
I A
l
a(T
here
is n
o si
ngle
ton
stru
ctur
e on
U if
(r.
.7(*
1.)
Exa
mpl
e 14
.A
n em
pty
set s
truc
ture
on
U=
16
4:
(The
re is
no
empt
y se
t str
uctu
re if
T7*
+)
Exa
mpl
e 15
.A
n In
volu
tion
on U
=1*
-,4.
, A.
hh
A
rrn
s
4e
a.
ET
C, E
TC
, ET
C,
...
78
CO
NV
EN
TIO
N :
An
arbi
trar
y st
ruct
ure
A. o
n T
...7
will
som
etim
es b
e re
pres
ente
dby
bc
Her
e, U
=5
Ad
6., c
., ,A
)e.
A s
truc
ture
A. o
n T
..7 is
said
to b
e ev
en (
odd)
if LT
con
tain
s an
eve
n(o
dd)
num
ber
of e
lem
ents
.
Tw
o st
ruct
ures
4, a
nd t
are
said
to b
e di
sjoi
nt
if th
eir
unde
rlyin
g se
ts a
redi
sjoi
nt.
Str
uctu
res
can
be s
tudi
edan
d cl
assi
fied
byco
llect
ing
them
into
SP
EC
IES
OF
ST
RU
CT
UR
ES
:
For
exam
ple,
we
can
cons
ider
the
spec
ies
ath
e sp
ecie
sA
the
spec
ies
the
spec
ies
ce.
the
spec
ies
a)th
e sp
ecie
sE
the
spec
ies
E *
van
the
spec
ies
Ew
ath
e sp
ecie
s S
the
spec
ies
L.th
e sp
ecie
s B
the
spec
ies
End
the
spec
ies
VI'
of a
llbi
colo
red
poly
gons
)th
e sp
ecie
s 21
4.0,
. of a
ll gr
aphs
,th
e sp
ecie
sof
all
conn
ecte
d gr
aphs
,
the
spec
ies
)(of
all
sing
leto
ns,
the
spec
ies
Iof
the
empt
y se
t,th
e em
pty
spec
ies
0(c
onta
inin
g no
str
uctu
re),
the
spec
ies
Inv
of a
llin
volu
tions
,
of a
ll tr
ees
(arb
res)
,
of a
ll ro
oted
tree
s(a
rbor
e5ce
nces
),
of a
ll cy
cles
,of
all
even
cyc
les,
of a
ll su
bset
s (o
f set
s),
of a
ll se
ts (
ense
mbl
es),
of a
ll ev
en s
ets,
of a
ll od
d se
ts,
of a
ll pe
rmut
atio
ns,
of a
ll lin
ear
orde
rs,
of a
ll pa
rtiti
ons,
of a
ll ed
ofun
ctio
ns,
the
spec
ies
Fof
all
F-
stru
ctur
es.
910
By
conv
entio
n, fo
r an
y fin
ite s
et tJ
we
writ
e
FLU
]:=
the
set o
f all
F-st
ruct
ures
on
U
Exa
mpl
e.If
F =
= th
e sp
ecie
s of
cyc
les,
then
Is a
41-
str
uctu
re
and
if"i
f=
rn,n
,1,,
11, t
hen
Cif
rn, )
1, 0
)131
AN
DirO
Ar't
ger
e't
"&tli
von
;:74.
."44
4171
44,5
r3
32
Fun
dam
enta
l pro
pert
ies
satis
fied
byth
e pr
ocee
ding
exa
mpl
e :
PR
OP
ER
TY
1.
For
eve
ry fi
nite
set
t7, t
he s
et47
E73
Is a
lway
s fin
ite.
PR
OP
ER
TY
2.
Eve
ry b
iject
ion
U =
rn,
rij
PV11
1 I
V =
14-
ha
c, 4
3in
duce
s an
othe
r bi
ject
ion,
CL
U)
44>
446
411,
24."
I0,
1=1
whi
ch r
elab
els
via
/3 e
ach
C-s
truc
ture
on
Uto
form
a c
orre
spon
ding
C -
stru
ctur
e on
V.
011
12
Not
e. T
wo
succ
essi
ve r
elab
ellin
gsvi
a A
and
Al
amou
nts
to a
sin
gle
rela
belli
ng v
ia A
'o /3
:
P4.)
flop
ou,
ctav
i )m
ei°
Cfp
le
cm=
CLA
."3
CD
DE
FIN
ITIO
N (
And
re J
oyal
,U
CIA
M,1
979)
A c
ombi
nato
rial s
peci
es is
aru
le F
whi
ch
1) g
ener
ates
, for
eac
h fin
ite s
etU
,an
othe
r fin
ite s
etF
LU),
2) g
ener
ates
, for
eac
h bi
ject
ion
FLU
)(3
*F
,an
othe
r bi
ject
ion
FE
V v)in
a c
oher
ent w
ay :
F[(
1° J
o Fr
iaj
F1(3
'0A
l
Any
a. I
n F
LU)
is c
alle
d a
F-s
truc
ture
on
U .
0 A
ny b
iject
ion
F [f
a] is
calle
d F
- re
labe
lling
via
p .
CO
NV
EN
TIO
N. A
n ar
bitr
ary
F -
stru
ctur
e A
L on
T..7
can
be r
epre
sent
edby
dia
gram
s lik
e
or
or
ET
C
Cou
ntin
g st
ruct
ures
of a
giv
ensp
ecie
s
The
pro
pert
ies
of r
elab
ellin
gsim
ply
that
the
num
ber
of F
-st
ruct
ures
on
t7de
pend
s on
ly o
n th
e nu
mbe
r of
elem
ents
of U
.
Tha
t is,
1 F
M I
depe
nds
only
on
l tj .
1314
Hen
ce, i
f we
wan
t to
sim
ply
coun
t F -
stru
ctur
es,
it is
suf
ficie
nt to
con
side
r on
ly th
e ca
ses
1.7
= n
=--
1..#
2.11
3n
= 0
,1,
2o .
.
DE
FIN
ITIO
N.
Let F
be
a sp
ecie
s an
d
=IF
te_1
1=
num
ber
of F
-st
ruct
ures
labe
lled
by1,
2,...
,ni.
The
gen
erat
ing
serie
s of
the
spec
ies
F is
00
F(x)
-7,
&I
n!=
F(x
) =
s xv
in
Exa
mpl
e 1.
S t
the
spec
ies
of p
erm
utat
ions
S(x
) =
E p
i!=
Ex"
nvo
0,1*
0I
Exa
mpl
e 2.
6:1
% th
e sp
ecie
s of
sub
sets
2"eZ
Xn3
9
Exa
mpl
e 3.
Eth
e sp
ecie
s of
set
s
EC
zJ=
=ex
fly.
ww
Exa
mpl
e 4.
;th
e sp
ecie
s of
eve
n se
ts
(x)
++
De*
+.=
cos
hN
OT
E. W
e w
ill s
ee th
at g
ener
atin
g se
ries
give
ris
eE
wen
to a
dyn
amic
con
nect
ion
betw
een
CO
MB
INA
TO
RIC
S a
nd A
NA
LYS
IS.
Exa
mpl
e 5.
Eye
dt
the
spec
ies
of o
dd s
ets
(1)=
+13
1-+
+ -
-=
sinl
nx
1516
Exa
mpl
e 6.
L th
e sp
ecie
s of
line
ar o
rder
s
L (
x) z
x"
Ex"
111.
.ePl
y°
Exa
mpl
e 7.
C ;
the
spec
ies
of c
ycle
s
C(x
)=
E X
" =
.0/n
(1%
ve,
)n
It I
Exa
mpl
e 8.
the
spec
ies
of e
ven
cycl
esIS
MS
c.(x
)= D
n.-0
14 m
zak
ir aq
nft
PI e
ven
itit,
I
Exa
mpl
e 9.
End
= th
e sp
ecie
s of
end
ofun
ctio
ns
End
(X
) -=
n"-F
rP1
3o
Exa
mpl
e 10
.X
:th
e sp
ecie
s of
sin
glet
ons
X(4
) =
++
Coi
fs=
Exa
mpl
e 11
.ith
m I
the
spec
ies
of g
raph
s
CZ
)"
Tc/
el a
CO
MB
ININ
G a
nd M
AN
IPU
LAT
ING
SP
EC
IES
We
know
how
toad
d, m
ultip
ly, s
ubst
itute
, and
diff
eren
tiate
serie
s to
obt
ain
othe
r se
ries
:
IfF
(x).
. Eib
.lir
kG
(30-E
i p, 24+
1.1
,,,,
INN
,
H (
x) =
Z 1
7.,
Inv
then
we
have
the
follo
win
g ta
ble
and OP
ER
AT
ION
CO
EF
FIC
IEN
T h
n
H(4
= F
(x)
+G
(x)
il =
5 +
5;,
1-1W
= F
(c)
- 6(
%)
4_10
1 ii.
h=as
-*
ti(7)
=F
(6(
2))
G(o
)= ie
r 0
=7"
,81
,ni
t6
...St
ink
3-4
4A"I
.
H (
x) =
ti-
F(x
)h
= 5
+,
Thi
s ta
ble
sugg
ests
cor
resp
ondi
ngco
mbi
nato
rial o
pera
tions
for
spec
ies
:
1718
DE
FIN
ITIO
N. F
rom
giv
en s
peci
es F
and
Got
her
spec
ies
can
be b
uilt
:
The
e4p
ecie
s F
+G
A (
F+
G)-
stru
ctur
esE
r(4
1.44
..tG
oph.
)
a 6-
stru
ctur
e
a F
-str
uctu
re
or
t\..)
The
spe
cies
Gan
ord
ered
pai
r
A (
F-G
)-
stru
ctur
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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