csm week 1: introductory cross-disciplinary seminar
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CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
CSM Week 1: Introductory Cross-Disciplinary Seminar
Combinatorial Enumeration
Dave WagnerUniversity of Waterloo
I. The Lagrange Implicit Function Theorem and Exponential Generating Functions
II. A Smorgasbord of Combinatorial Identities
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
3. There are rooted trees (two ways)1nn
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
3. There are rooted trees (two ways)
4. Combinatorial proof (sketch) of LIFT
1nn
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
3. There are rooted trees (two ways)
4. Combinatorial proof (sketch) of LIFT
5. Nested set systems
1nn
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
3. There are rooted trees (two ways)
4. Combinatorial proof (sketch) of LIFT
5. Nested set systems
6. Multivariate Lagrange
1nn
I. LIFT and Exponential Generating Functions
1. The Lagrange Implicit Function Theorem
2. Exponential Generating Functions
3. There are rooted trees (two ways)
4. Combinatorial proof (sketch) of LIFT
5. Nested set systems
6. Multivariate Lagrange
1nn
The human mind has never invented a labor-saving device equal to algebra.
-- J. Willard Gibbs
1. The Lagrange Implicit Function Theorem
K: a commutative ring that contains the rational numbers.
F(u) and G(u): formal power series in K[[u]]:
Assume that
n
nnufuF
0
)( n
nnuguG
0
)(
.00 g
1. The Lagrange Implicit Function Theorem
K: a commutative ring that contains the rational numbers.
F(u) and G(u): formal power series in K[[u]]:
Assume that
(a) There is a unique formal power series R(x) in K[[x]]
such that
n
nnufuF
0
)( n
nnuguG
0
)(
.00 g
)).(()( xRxGxR
1. The Lagrange Implicit Function Theorem
(b) For this formal power series with
the constant term is zero:
))(()( xRxGxR
0
)(n
nnxrxR .00 r
1. The Lagrange Implicit Function Theorem
(b) For this formal power series with
the constant term is zero:
For all n>=1 the coefficient of x^n in
F(R(x)) is
))(()( xRxGxR
0
)(n
nnxrxR .00 r
.)()(][1
))((][ 1 nnn uGuFun
xRFx
1. The Lagrange Implicit Function Theorem
Proofs:
(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of
convergence]
1. The Lagrange Implicit Function Theorem
Proofs:
(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of
convergence]
(ii) Algebraic (formal calculus, formal residue operator)
[requires g_0 to be invertible in K]
1. The Lagrange Implicit Function Theorem
Proofs:
(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of
convergence]
(ii) Algebraic (formal calculus, formal residue operator)
[requires g_0 to be invertible in K]
(iii) Combinatorial (bijective correspondence).)()(][1
))((][ 1 nnn uGuFun
xRFx
1. The Lagrange Implicit Function Theorem
Proofs:
(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of
convergence]
(ii) Algebraic (formal calculus, formal residue operator)
[requires g_0 to be invertible in K]
(iii) Combinatorial (bijective correspondence).)()(][))((][ 1 nnn uGuFuxRFxn
1. The Lagrange Implicit Function Theorem
Proofs:
(i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of
convergence]
(ii) Algebraic (formal calculus, formal residue operator)
[requires g_0 to be invertible in K]
(iii) Combinatorial (bijective correspondence) .)()(][!))((][! 1 nnn uGuFunxRFxnn
2. Exponential Generating Functions
A class of structures associates to each
finite set another finite set -- this is
the set of A-type structures supported on the set X.
A
X XA
2. Exponential Generating Functions
A class of structures associates to each
finite set another finite set -- this is
the set of A-type structures supported on the set X.
Simplified notation:
A
X XA
.},...,2,1{ nn AA
2. Exponential Generating Functions
A class of structures associates to each
finite set another finite set -- this is
the set of A-type structures supported on the set X.
Simplified notation:
Exponential generating function:
A
X XA
.},...,2,1{ nn AA
!
#)(0 n
xxA
n
nn
A
2. Exponential Generating Functions
Minimal requirements on a class of structures:
* depends only on
* If then
XA# X#
YX YX AA
2. Exponential Generating Functions
Example: the class of (simple) graphs
is the set of graphs with vertex-set
Exponential generating function
(no particularly useful formula)
G
XG X
2
#
2X
XG#
!2)(
0
2
n
xxG
n
n
n
2. Exponential Generating Functions
Example: the class of endofunctions
is the set of all functions
Exponential generating function
(no particularly useful formula)
X XX :
XX X ###
!)(
0 n
xnx
n
n
n
2. Exponential Generating Functions
Example: the class of permutations
is the set of permutations on the set
Exponential generating function
S
XS X
)!(# XX S#
xn
xnxS
n
n
1
1
!!)(
0
2. Exponential Generating Functions
Example: the class of cyclic permutations
is the set of cyclic perm.s on the set
Exponential generating function
C
XC X
XX
XX )!1(#
0C#
xn
x
n
xnxC
n
nn
n 1
1log
!)!1()(
11
2. Exponential Generating Functions
Example: the class of (finite) sets (“ensembles”)
is the set of ways in which is a set.
Exponential generating function
E
}{XX E X
1XE#
)exp(!
1)(1
xn
xxE
n
n
2. Exponential Generating Functions
Example: the class of sets of size k
is the set of ways in which
is a k-element set.
Exponential generating function
Especially important: the case k=1 of singletons….
has exp.gen.fn x.
)(kE
kX
kXXkX #
#}{)(E X
!)()(
k
xxE
kk
)1(EX
2. Exponential Generating Functions
Notice that
xx 1
1logexp
1
1
2. Exponential Generating Functions
Notice that
That is… )).(()( xCExS
xx 1
1logexp
1
1
2. Exponential Generating Functions
Notice that
That is…
This suggests a relation among classes: E[C].S
)).(()( xCExS
xx 1
1logexp
1
1
2. Exponential Generating Functions
Notice that
That is…
This suggests a relation among classes:
A permutation is equivalent to a (finite unordered)set of (pairwise disjoint) cyclic permutations.
E[C].S
)).(()( xCExS
xx 1
1logexp
1
1
2. Exponential Generating Functions
X
2. Exponential Generating Functions
A permutation is equivalent to a (finite unordered)set of (pairwise disjoint) cyclic permutations.
X
2. Exponential Generating Functions
X
xx 1
1logexp
1
1
2. Exponential Generating Functions
An endofunction is equivalent to a set of disjointconnected endofunctions.
X
2. Exponential Generating Functions
A connected endofunction is equivalent to a cyclic permutation of rooted trees.
X
2. Exponential Generating Functions
A connected endofunction is equivalent to a cyclic permutation of rooted trees.
X
2. Exponential Generating Functions
A rooted tree is equivalent to a root vertex and aset of disjoint rooted (sub-)trees.
X
2. Exponential Generating Functions
A rooted tree is equivalent to a root vertex and aset of disjoint rooted (sub-)trees.
X
2. Exponential Generating Functions
The Exponential/Logarithmic Formula
For classes A and B,
If every B-structure can be decomposed uniquely as a
finite set of pairwise disjoint A-structures, then
)(exp)( xAxB
2. Exponential Generating Functions
The Exponential/Logarithmic Formula
For classes A and B,
If every B-structure can be decomposed uniquely as a
finite set of pairwise disjoint A-structures, then
and hence
)(exp)( xAxB
)(log)( xBxA
2. Exponential Generating Functions
Example: Let Q be the class of connected graphs.
2. Exponential Generating Functions
Example: Let Q be the class of connected graphs.
Since it follows that
!2log)(
0
2
n
xxQ
n
n
n
E[Q]G
2. Exponential Generating Functions
Example: Let Q be the class of connected graphs.Since it follows that
More informatively,
records the number of edges in the exponent of y.
!2log)(
0
2
n
xxQ
n
n
n
0
2
0
)(#
!)1(log
!),(
n
nnn
n Q
E
n
xy
n
xyyxQ
n
E[Q]G
2. Exponential Generating Functions
The Compositional Formula
For classes A, B, and J:
If every B-structure can be decomposed uniquely as a
finite set Y of pairwise disjoint A-structures, together
with a J-structure on Y, then )()( xAJxB
2. Exponential Generating Functions
Example: Let K be the class of connectedendofunctions. Let R be the class of rooted trees.
2. Exponential Generating Functions
Example: Let K be the class of connectedendofunctions. Let R be the class of rooted trees.
Since it follows that C[R]K
)(1
1log))(()(
xRxRCxK
2. Exponential Generating Functions
Example: Let K be the class of connectedendofunctions. Let R be the class of rooted trees.
Since it follows that
Since it follows that
)(1
1)(exp)(
xRxKx
C[R]K
E[K]
)(1
1log))(()(
xRxRCxK
2. Exponential Generating Functions
Sum of classes A and B
An structure on X iseither a red A-structure or a green B-structure on X.
X
-BA
2. Exponential Generating Functions
An structure on X
X
-RS
2. Exponential Generating Functions
An structure on X
X
-RS
2. Exponential Generating Functions
Sum of classes A and BThe exp.gen.fn of is
X
BA
)()( xBxA
2. Exponential Generating Functions
Product of classes A and B
An structure on X is an A-structure on Sand a B-structure on X\S (for some subset S of X).
X
-BA*
S SX \
2. Exponential Generating Functions
An structure on X
X
-RS*
S SX \
2. Exponential Generating Functions
Product of classes A and BThe exp.gen.fn of is
X
BA*
S SX \
)()( xBxA
3. Counting Rooted Trees
A rooted tree is equivalent to a root vertex and aset of disjoint rooted (sub-)trees.
X
3. Counting Rooted Trees
X
E[R] XR *
3. Counting Rooted Trees
E[R] XR *
)(exp)( xRxxR
From
we deduce that
3. Counting Rooted Trees
E[R] XR *
)(exp)( xRxxR
From
we deduce that
LIFT applies with F(u)=u and G(u)=exp(u):
3. Counting Rooted Trees
!!
)(][
1)exp(][
1)(][
1
0
11
n
n
k
nuu
nuu
nxRx
n
k
knnnn
E[R] XR *
)(exp)( xRxxR
From
we deduce that
LIFT applies with F(u)=u and G(u)=exp(u):
3. Counting Rooted Trees
!!
)(][
1)exp(][
1)(][
1
0
11
n
n
k
nuu
nuu
nxRx
n
k
knnnn
E[R] XR *
)(exp)( xRxxR
From
we deduce that
LIFT applies with F(u)=u and G(u)=exp(u):
Therefore .)(][!# 1 nnn nxRxnR
5. Nested Set Systems
A nested set system is a pair
in which X is a finite set and is a set of subsets of X
such that
if and then either
or or .
Let N be the class of nested set systems. What is #N_n?
),( X
A B
BA AB BA
5. Nested Set Systems
X
A nested set system with vertex-set X.
5. Nested Set Systems
Let N be the class of nested set systems. What is #N_n?
5. Nested Set Systems
Let N be the class of nested set systems. What is #N_n?
We’ll use the bivariate generating function
!),(
0 ),(
#
n
xyyxN
n
n X n
N
5. Nested Set Systems
Let N be the class of nested set systems. What is #N_n?
We’ll use the bivariate generating function
This is an exp.gen.fn in the indeterminate x
and records in the exponent of y.
!),(
0 ),(
#
n
xyyxN
n
n X n
N
#
5. Nested Set Systems
X
A nested set system is proper if it does not containany sets of size zero or one.
Let M be the class of proper nested set systems
5. Nested Set Systems
X
A proper nested set system
5. Nested Set Systems
X
),)1(()1(),( yxyMyyxN
5. Nested Set Systems
X
),)1(()1(),( yxyMyyxN
5. Nested Set Systems
X
A proper nested set system is equivalent to a set of disjoint
blobs – each blob is a singleton or a “cell”.
5. Nested Set Systems
X
A cell is a proper nested set systemfor which --
Let Q be the class of cells.
),( XX
5. Nested Set Systems
X
A proper nested set system is equivalent to a set of disjoint
blobs – each blob is a singleton or a “cell”.
5. Nested Set Systems
X
Q]E[XM
5. Nested Set Systems
X
Q]E[XM
),(exp),( yxQxyxM
The “protoplasm” of a cell is a proper nested set system
that is not empty, not a singleton, and not a cell.
5. Nested Set Systems
X
The “protoplasm” of a cell is a proper nested set system
that is not empty, not a singleton, and not a cell.
5. Nested Set Systems
X
5. Nested Set Systems
X
QXE\MQ )0(
5. Nested Set Systems
X
QXE\MQ )0(
),(1),(),( yxQxyxMyyxQ
5. Nested Set Systems
X
QXE\MQ )0(
),(1),(),( yxQxyxMyyxQ
5. Nested Set Systems
),)1(()1(),( yxyMyyxN
QxM exp
yQyxyyMQ
5. Nested Set Systems
),)1(()1(),( yxyMyyxN
QxM exp
yQyxyyMQ
xMy
yQ
11
5. Nested Set Systems
),)1(()1(),( yxyMyyxN
QxM exp
yQyxyyMQ
xMy
yQ
11
xM
y
yxM 11
exp
5. Nested Set Systems
xM
y
yxM 11
exp
5. Nested Set Systems
xM
y
yxM 11
exp
y
yM
y
yx
y
y
y
yM
1exp
1exp
11
5. Nested Set Systems
xM
y
yxM 11
exp
y
yM
y
yx
y
y
y
yM
1exp
1exp
11
Let and
y
yx
y
yz
1exp
1y
yMR
1
5. Nested Set Systems
)exp(RzR
5. Nested Set Systems
)exp(RzR
1
1
!k
kk
k
zkR
5. Nested Set Systems
)exp(RzR
1
1
!k
kk
k
zkR
k
k
k
y
yx
y
y
k
k
y
yM
1exp
1!
1
1
1
5. Nested Set Systems
)exp(RzR
1
1
!k
kk
k
zkR
k
k
k
y
yx
y
y
k
k
y
yM
1exp
1!
1
1
1
k
k
k
y
yxy
y
y
k
k
y
yyxN
1
)1(exp
1!
)1(),(
1
12
5. Nested Set Systems
2
1exp
2!4)1,(
1
1
xkk
kxN
kk
k
5. Nested Set Systems
2
1exp
2!4)1,(
1
1
xkk
kxN
kk
k
1
2/
1
2!4)1,(][!#k
kk
knn
n ek
kxNxnN
Therefore, the number of nested set systems on the
vertex-set {1,2,…,n} is
• n, ~ #N_n (up to k = 500) (k = 500 term of the series)
• 0, 2.000000000000000000000000000000000000000, .4083243888661365954428680604080286918931e-46• 1, 3.999999999999999999999999999999999999997, .2041621944330682977214340302040143459465e-43• 2, 16.00000000000000000000000000000000000000, .1020810972165341488607170151020071729732e-40• 3, 127.9999999999999999999999999999999999998, .5104054860826707443035850755100358648663e-38• 4, 1663.999999999999999999999999999999999951, .2552027430413353721517925377550179324331e-35• 5, 30207.99999999999999999999999999999997499, .1276013715206676860758962688775089662165e-32• 6, 704511.9999999999999999999999999999873828, .6380068576033384303794813443875448310827e-30• 7, 20074495.99999999999999999999999999361776, .3190034288016692151897406721937724155414e-27• 8, 675872767.9999999999999999999999967706124, .1595017144008346075948703360968862077707e-24• 9, 26253131775.99999999999999999999836574318, .7975085720041730379743516804844310388536e-22• 10, 1155636527103.999999999999999999172874000, .3987542860020865189871758402422155194268e-19• 11, 56851643236351.99999999999999958132597320, .1993771430010432594935879201211077597134e-16• 12, 3091106738733055.999999999999788049473799, .9968857150052162974679396006055387985669e-14• 13, 184069292705185791.9999999998926879966856, .4984428575026081487339698003027693992835e-11• 14, 11913835525552734207.99999994566012222874, .2492214287513040743669849001513846996417e-8• 15, 832795579840760643583.9999724801012619416, .1246107143756520371834924500756923498209e-5• 16, 62525006404716521848831.98606091920623040, .6230535718782601859174622503784617491043e-3• 17, 5017971241212451282223096.938744559857556, .3115267859391300929587311251892308745523• 18, 428697615765805738749850118.4015658433400, 155.7633929695650464793655625946154372761• 19, 38844089835957753021198521986.64691355734, 77881.69648478252323968278129730771863803
5. Nested Set Systems
References
A. JoyalUne theorie combinatoire des series formellesAdv. in Math. 42 (1981), 1-82.
I.P. Goulden, D.M. Jackson“Combinatorial Enumeration”John Wiley & Sons, New York, 1983.
F. Bergeron, G. Labelle, P. Leroux“Combinatorial Species and Tree-like Structures”Cambridge U.P., Cambridge, 1998.
R.P. Stanley“Enumerative Combinatorics, volume II”Cambridge U.P., Cambridge, 1999.
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