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Exponential Generating Functions
The Exponential Formula
Topics in Algebraic Combinatorics
Oliver Zhang • Proof School • 9th Grade, Block 1, 2016
The Exponential Formula
Exponential Generating Functions
What is an EGF?
An EGF or Exponential Generating Function
is a generating function over a factorial.
For Example:
If Aegf←→ {an}∞0 then
A =
∞∑n=0
ann!xn
The Exponential Formula
Exponential Generating Functions
What is an EGF?
An EGF or Exponential Generating Function
is a generating function over a factorial.
For Example:
If Aegf←→ {an}∞0 then
A =
∞∑n=0
ann!xn
The Exponential Formula
Exponential Generating Functions
What is an EGF?
An EGF or Exponential Generating Function
is a generating function over a factorial.
For Example:
If Aegf←→ {an}∞0 then
A =
∞∑n=0
ann!xn
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs
Given two EGFs A and B, what EGF would
A ·B be?
In other words, if( ∞∑n=0
ann!x
n
)( ∞∑n=0
bnn!x
n
)=∞∑n=0
cnn!x
n
then what would cn be in terms of an and bn?
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs
Given two EGFs A and B, what EGF would
A ·B be?
In other words, if( ∞∑n=0
ann!x
n
)( ∞∑n=0
bnn!x
n
)=∞∑n=0
cnn!x
n
then what would cn be in terms of an and bn?
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs, cont.
Let’s take an example. To calculate c44! , we
simply take the sum
a0b40!4! +
a1b31!3! +
a2b22!2! +
a3b13!1! +
a4b04!0!
Which is equal to
a0b44! +
4!1!3!a1b3
4! +4!2!2!a2b2
4! +4!3!1!a3b1
4! + a4b04!
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs, cont.
Let’s take an example. To calculate c44! , we
simply take the sum
a0b40!4! +
a1b31!3! +
a2b22!2! +
a3b13!1! +
a4b04!0!
Which is equal to
a0b44! +
4!1!3!a1b3
4! +4!2!2!a2b2
4! +4!3!1!a3b1
4! + a4b04!
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs, cont.
Therefore, c4 is equal to the sum:(40
)a0b4 +
(41
)a1b3 +
(42
)a2b2 +
(43
)a3b1 +
(44
)a4b0
Generally, cn will be equal ton∑
m=0
(nm
)ambn−m
The Exponential Formula
Exponential Generating Functions
Multiplying EGFs, cont.
Therefore, c4 is equal to the sum:(40
)a0b4 +
(41
)a1b3 +
(42
)a2b2 +
(43
)a3b1 +
(44
)a4b0
Generally, cn will be equal ton∑
m=0
(nm
)ambn−m
The Exponential Formula
Exponential Generating Functions
The Exponential Function
Just as
11−x ←→ 1 + x + x2 + ...
What f (x)egf←→ 1 + x
1! +x2
2! + ...?
The Exponential Formula
Exponential Generating Functions
The Exponential Function
Correct!
exegf←→ 1 + x
1! +x2
2! + ...
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph:
a set of vertices and relationsbetween the vertices.
A connected component of a graph: a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture: a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph: a set of vertices and relationsbetween the vertices.
A connected component of a graph: a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture: a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph: a set of vertices and relationsbetween the vertices.
A connected component of a graph:
a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture: a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph: a set of vertices and relationsbetween the vertices.
A connected component of a graph: a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture: a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph: a set of vertices and relationsbetween the vertices.
A connected component of a graph: a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture:
a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
A few definitions:
A graph: a set of vertices and relationsbetween the vertices.
A connected component of a graph: a set ofvertices within a graph such that any twovertices are connected to each other by paths,and which is connected to no additional verticesin the supergraph.
A picture: a connected graph such that the setof vertices is the set {1, 2, ..., n}, where n is thenumber of vertices of the graph.
The Exponential Formula
Exponential Generating Functions
Graph Theory
Note: These two pictures are considered
distinct because the labeling of their vertices
are different.
The Exponential Formula
Exponential Generating Functions
Million Dollar Question
Our Question is:
How many ways are there to build a graph withn vertices and k connected components?
The Exponential Formula
Exponential Generating Functions
Million Dollar Question
Our Question is:
How many ways are there to build a graph withn vertices and k connected components?
The Exponential Formula
Exponential Generating Functions
Poker Playing: The Basic Unit
A card C(S, p):
A non-empty ’label set’ S
A ’picture’ p
|S| = number of vertices in p.
Each card represents a connected component
of a graph.
The Exponential Formula
Exponential Generating Functions
Poker Playing: The Basic Unit
A card C(S, p):A non-empty ’label set’ S
A ’picture’ p
|S| = number of vertices in p.
Each card represents a connected component
of a graph.
The Exponential Formula
Exponential Generating Functions
Poker Playing: The Basic Unit
A card C(S, p):A non-empty ’label set’ S
A ’picture’ p
|S| = number of vertices in p.
Each card represents a connected component
of a graph.
The Exponential Formula
Exponential Generating Functions
Poker Playing: The Basic Unit
A card C(S, p):A non-empty ’label set’ S
A ’picture’ p
|S| = number of vertices in p.
Each card represents a connected component
of a graph.
The Exponential Formula
Exponential Generating Functions
Poker Playing: The Basic Unit
A card C(S, p):A non-empty ’label set’ S
A ’picture’ p
|S| = number of vertices in p.
Each card represents a connected component
of a graph.
The Exponential Formula
Exponential Generating Functions
Card Example
A card is called standard if the label set S is
the set {1, 2, ..., n} for some n.
The Exponential Formula
Exponential Generating Functions
Card Example
A card is called standard if the label set S is
the set {1, 2, ..., n} for some n.
The Exponential Formula
Exponential Generating Functions
Card Example
A card is called standard if the label set S is
the set {1, 2, ..., n} for some n.
The Exponential Formula
Exponential Generating Functions
Card Example
A card is called standard if the label set S is
the set {1, 2, ..., n} for some n.
The Exponential Formula
Exponential Generating Functions
Card Example
A card is called standard if the label set S is
the set {1, 2, ..., n} for some n.The Exponential Formula
Exponential Generating Functions
Hands
If a connected component can be represented
by a card, what would a graph be?
A hand H is a set of cards whose label sets
form a partition of {1, 2, ..., n} for some n.
A hand is usually created with cards from a
specified Exponential Family, we’ll get to
that in a bit
The Exponential Formula
Exponential Generating Functions
Hands
If a connected component can be represented
by a card, what would a graph be?
A hand H is a set of cards whose label sets
form a partition of {1, 2, ..., n} for some n.
A hand is usually created with cards from a
specified Exponential Family, we’ll get to
that in a bit
The Exponential Formula
Exponential Generating Functions
Hands
If a connected component can be represented
by a card, what would a graph be?
A hand H is a set of cards whose label sets
form a partition of {1, 2, ..., n} for some n.
A hand is usually created with cards from a
specified Exponential Family,
we’ll get to
that in a bit
The Exponential Formula
Exponential Generating Functions
Hands
If a connected component can be represented
by a card, what would a graph be?
A hand H is a set of cards whose label sets
form a partition of {1, 2, ..., n} for some n.
A hand is usually created with cards from a
specified Exponential Family, we’ll get to
that in a bit
The Exponential Formula
Exponential Generating Functions
Hand Example
The weight of a card is the size of the label
set. The weight of a hand is the sum of the
weight of the cards.
The Exponential Formula
Exponential Generating Functions
Hand Example
The weight of a card is the size of the label
set. The weight of a hand is the sum of the
weight of the cards.
The Exponential Formula
Exponential Generating Functions
Hand Example
The weight of a card is the size of the label
set. The weight of a hand is the sum of the
weight of the cards.
The Exponential Formula
Exponential Generating Functions
Hand Example
The weight of a card is the size of the label
set. The weight of a hand is the sum of the
weight of the cards.
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be? 3
What would h(2,3) be? 0
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be?
3
What would h(2,3) be? 0
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be? 3
What would h(2,3) be? 0
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be? 3
What would h(2,3) be?
0
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be? 3
What would h(2,3) be? 0
The Exponential Formula
Exponential Generating Functions
Notation!
We denote h(n, k) to be the number of
hands with weight n and k cards.
What would h(3,2) be? 3
What would h(2,3) be? 0
The Exponential Formula
Exponential Generating Functions
Million Dollar Question Revisited
Our Question:
How many ways are there to build a graph withn vertices and k connected components?
Can be rephrased to:
How many hands are there with weight n and kcards?
or what is h(n, k)?
The Exponential Formula
Exponential Generating Functions
Million Dollar Question Revisited
Our Question:
How many ways are there to build a graph withn vertices and k connected components?
Can be rephrased to:
How many hands are there with weight n and kcards?
or what is h(n, k)?
The Exponential Formula
Exponential Generating Functions
Million Dollar Question Revisited
Our Question:
How many ways are there to build a graph withn vertices and k connected components?
Can be rephrased to:
How many hands are there with weight n and kcards?
or what is h(n, k)?
The Exponential Formula
Exponential Generating Functions
Decks
A deck Dn is a finite set of standard cards
whose weights are all n and whose pictures
are all different.
The Exponential Formula
Exponential Generating Functions
Decks
A deck Dn is a finite set of standard cards
whose weights are all n and whose pictures
are all different.
The Exponential Formula
Exponential Generating Functions
Exponential Families
An exponential family [EF] F is defined to
be a collection of decks with weights 1, 2, ...
In an exponential family, let dn be defined as
the number of cards in deck Dn
The Exponential Formula
Exponential Generating Functions
Exponential Families
An exponential family [EF] F is defined to
be a collection of decks with weights 1, 2, ...
In an exponential family, let dn be defined as
the number of cards in deck Dn
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be? 1
What would d′2 be? 1
What would d′3 be? 4
What about d′4? 38
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be?
1
What would d′2 be? 1
What would d′3 be? 4
What about d′4? 38
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be? 1
What would d′2 be?
1
What would d′3 be? 4
What about d′4? 38
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be? 1
What would d′2 be? 1
What would d′3 be?
4
What about d′4? 38
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be? 1
What would d′2 be? 1
What would d′3 be? 4
What about d′4?
38
The Exponential Formula
Exponential Generating Functions
Exponential Families Example
If D′n was the deck with all cards of weight
n, then
What would d′1 be? 1
What would d′2 be? 1
What would d′3 be? 4
What about d′4? 38
The Exponential Formula
Exponential Generating Functions
Generating Functions pt. 1
We can finally start building generating
functions! Yay!
We introduce the 2-variable Hand
Enumerator
H(x, y) =∑n,k≥0
h(n, k)xn
n!yk
This generating function is a half-blood.
The Exponential Formula
Exponential Generating Functions
Generating Functions pt. 1
We can finally start building generating
functions! Yay!
We introduce the 2-variable Hand
Enumerator
H(x, y) =∑n,k≥0
h(n, k)xn
n!yk
This generating function is a half-blood.
The Exponential Formula
Exponential Generating Functions
Generating Functions pt. 1
We can finally start building generating
functions! Yay!
We introduce the 2-variable Hand
Enumerator
H(x, y) =∑n,k≥0
h(n, k)xn
n!yk
This generating function is a half-blood.
The Exponential Formula
Exponential Generating Functions
Generating Functions pt. 2
The deck enumerator is the egf of the
sequence {dn}∞1 and is denoted as D(x).
The Exponential Formula
Exponential Generating Functions
Merging Families
Given two EFs F ′ and F ′′ with no shared
pictures,
we can merge them together to
create a new exponential family F such that
For any D′n and D′′n in F ′ and F ′′
respectively, Dn in F is the union of the card
sets of D′n and D′′n.
The Exponential Formula
Exponential Generating Functions
Merging Families
Given two EFs F ′ and F ′′ with no shared
pictures, we can merge them together to
create a new exponential family F such that
For any D′n and D′′n in F ′ and F ′′
respectively, Dn in F is the union of the card
sets of D′n and D′′n.
The Exponential Formula
Exponential Generating Functions
Merging Families
Given two EFs F ′ and F ′′ with no shared
pictures, we can merge them together to
create a new exponential family F such that
For any D′n and D′′n in F ′ and F ′′
respectively, Dn in F is the union of the card
sets of D′n and D′′n.
The Exponential Formula
Exponential Generating Functions
A Quick Lemma
Lemma (Label Counting)
Let F , F ′, F ′′ be three exponential families
and let H(x, y), H ′(x, y), H ′′(x, y) be their
respective 2-variable hand enumerators. If
F = F ′ ⊗ F ′′
Then
H(x,y) = H’(x,y)H”(x,y)
The Exponential Formula
Exponential Generating Functions
A Quick Lemma
Lemma (Label Counting)
Let F , F ′, F ′′ be three exponential families
and let H(x, y), H ′(x, y), H ′′(x, y) be their
respective 2-variable hand enumerators. If
F = F ′ ⊗ F ′′
Then
H(x,y) = H’(x,y)H”(x,y)
The Exponential Formula
Exponential Generating Functions
A Quick Lemma
Lemma (Label Counting)
Let F , F ′, F ′′ be three exponential families
and let H(x, y), H ′(x, y), H ′′(x, y) be their
respective 2-variable hand enumerators. If
F = F ′ ⊗ F ′′
Then
H(x,y) = H’(x,y)H”(x,y)
The Exponential Formula
Exponential Generating Functions
The Initial Case
Given a fixed r, let all decks besides Dr be
empty. Additionally, let Dr only contain one
card with r vertices.
h(n, k) for this deck is zero unless n can be
represented by kr for some integer k.
But then how many hands are there?
The Exponential Formula
Exponential Generating Functions
The Initial Case
Given a fixed r, let all decks besides Dr be
empty. Additionally, let Dr only contain one
card with r vertices.
h(n, k) for this deck is zero unless n can be
represented by kr for some integer k.
But then how many hands are there?
The Exponential Formula
Exponential Generating Functions
The Initial Case
The first card labels can be picked in(nr
)ways, the second card in
(n−rr
)ways, ... the
last card can be picked in(n−(k−1)r
r
)= 1 way.
Additionally, the order of the k cards doesn’t
matter so we divide out another k!.
h(kr, k) = 1k!
n!r!k
The Exponential Formula
Exponential Generating Functions
The Initial Case
The first card labels can be picked in(nr
)ways, the second card in
(n−rr
)ways, ... the
last card can be picked in(n−(k−1)r
r
)= 1 way.
Additionally, the order of the k cards doesn’t
matter so we divide out another k!.
h(kr, k) = 1k!
n!r!k
The Exponential Formula
Exponential Generating Functions
The Initial Case
The first card labels can be picked in(nr
)ways, the second card in
(n−rr
)ways, ... the
last card can be picked in(n−(k−1)r
r
)= 1 way.
Additionally, the order of the k cards doesn’t
matter so we divide out another k!.
h(kr, k) = 1k!
n!r!k
The Exponential Formula
Exponential Generating Functions
The Initial Case
If h(kr, k) = 1k!
n!r!k
then the hand enumerator
of this family is
H(x, y) =∑n,k
h(n, k)xnyk/n!
=∑k
xkryk
k!r!k
=∑k
(xryr!
)k/k!
= exp{yxr
r! }
The Exponential Formula
Exponential Generating Functions
The Initial Case
If h(kr, k) = 1k!
n!r!k
then the hand enumerator
of this family is
H(x, y) =∑n,k
h(n, k)xnyk/n!
=∑k
xkryk
k!r!k
=∑k
(xryr!
)k/k!
= exp{yxr
r! }
The Exponential Formula
Exponential Generating Functions
The Initial Case
If h(kr, k) = 1k!
n!r!k
then the hand enumerator
of this family is
H(x, y) =∑n,k
h(n, k)xnyk/n!
=∑k
xkryk
k!r!k
=∑k
(xryr!
)k/k!
= exp{yxr
r! }
The Exponential Formula
Exponential Generating Functions
The Initial Case
If h(kr, k) = 1k!
n!r!k
then the hand enumerator
of this family is
H(x, y) =∑n,k
h(n, k)xnyk/n!
=∑k
xkryk
k!r!k
=∑k
(xryr!
)k/k!
= exp{yxr
r! }
The Exponential Formula
Exponential Generating Functions
The Initial Case
If h(kr, k) = 1k!
n!r!k
then the hand enumerator
of this family is
H(x, y) =∑n,k
h(n, k)xnyk/n!
=∑k
xkryk
k!r!k
=∑k
(xryr!
)k/k!
= exp{yxr
r! }
The Exponential Formula
Exponential Generating Functions
Induction
Given an exponential family F and positive
integer r such that
every deck besides Dr is
empty, we claim its hand enumerator is:
H(x, y) = exp{ydrxr
r! }
The Exponential Formula
Exponential Generating Functions
Induction
Given an exponential family F and positive
integer r such that every deck besides Dr is
empty,
we claim its hand enumerator is:
H(x, y) = exp{ydrxr
r! }
The Exponential Formula
Exponential Generating Functions
Induction
Given an exponential family F and positive
integer r such that every deck besides Dr is
empty, we claim its hand enumerator is:
H(x, y) = exp{ydrxr
r! }
The Exponential Formula
Exponential Generating Functions
Induction
Suppose the claim is true for
dr = 1, 2, ...m− 1 and let F have m cards
in its rth deck.
Then F is the result of
merging a family with m− 1 cards in the rth
deck and a family with 1 card in that deck.
exp{y(m− 1)xr/r!}exp{yxr/r!}= exp{ymxr/r!}
The Exponential Formula
Exponential Generating Functions
Induction
Suppose the claim is true for
dr = 1, 2, ...m− 1 and let F have m cards
in its rth deck. Then F is the result of
merging a family with m− 1 cards in the rth
deck and a family with 1 card in that deck.
exp{y(m− 1)xr/r!}exp{yxr/r!}= exp{ymxr/r!}
The Exponential Formula
Exponential Generating Functions
Induction
Suppose the claim is true for
dr = 1, 2, ...m− 1 and let F have m cards
in its rth deck. Then F is the result of
merging a family with m− 1 cards in the rth
deck and a family with 1 card in that deck.
exp{y(m− 1)xr/r!}
exp{yxr/r!}= exp{ymxr/r!}
The Exponential Formula
Exponential Generating Functions
Induction
Suppose the claim is true for
dr = 1, 2, ...m− 1 and let F have m cards
in its rth deck. Then F is the result of
merging a family with m− 1 cards in the rth
deck and a family with 1 card in that deck.
exp{y(m− 1)xr/r!}exp{yxr/r!}
= exp{ymxr/r!}
The Exponential Formula
Exponential Generating Functions
Induction
Suppose the claim is true for
dr = 1, 2, ...m− 1 and let F have m cards
in its rth deck. Then F is the result of
merging a family with m− 1 cards in the rth
deck and a family with 1 card in that deck.
exp{y(m− 1)xr/r!}exp{yxr/r!}= exp{ymxr/r!}
The Exponential Formula
Exponential Generating Functions
Multi-deck Induction
Theorem (The Exponential Formula)Let F be an exponential family whose deck andhand enumerators are D(x) and H(x, y).Then
H(x, y) = eyD(x)
andh(n, k) =
[xn
n!
] {D(x)k
k!
}
The Exponential Formula
Exponential Generating Functions
Multi-deck Induction
By merging a sequence of decks D1, D2, ...
we can obtain any exponential family F .
= exp{
yd1x1
1!
}exp{
yd2x2
2!
}...
= exp{
yd1x1
1! + yd2x2
2! + ...}
= eyD(x)
The Exponential Formula
Exponential Generating Functions
Multi-deck Induction
By merging a sequence of decks D1, D2, ...
we can obtain any exponential family F .
= exp{
yd1x1
1!
}exp{
yd2x2
2!
}...
= exp{
yd1x1
1! + yd2x2
2! + ...}
= eyD(x)
The Exponential Formula
Exponential Generating Functions
Multi-deck Induction
By merging a sequence of decks D1, D2, ...
we can obtain any exponential family F .
= exp{
yd1x1
1!
}exp{
yd2x2
2!
}...
= exp{
yd1x1
1! + yd2x2
2! + ...}
= eyD(x)
The Exponential Formula
Exponential Generating Functions
Multi-deck Induction
By merging a sequence of decks D1, D2, ...
we can obtain any exponential family F .
= exp{
yd1x1
1!
}exp{
yd2x2
2!
}...
= exp{
yd1x1
1! + yd2x2
2! + ...}
= eyD(x)
The Exponential Formula
Exponential Generating Functions
Solving the question
The number of ways to build a graph with nvertices and k connected components?is just
h(n, k) = H(x, y)[xn
n! yk]
= eyD(x)[xn
n! yk]
=[xn
n! yk] {ykD(x)k
k!
}∞0
=[xn
n!
] {D(x)k
k!
}∞0
The Exponential Formula
Exponential Generating Functions
Solving the question
The number of ways to build a graph with nvertices and k connected components?is just
h(n, k) = H(x, y)[xn
n! yk]
= eyD(x)[xn
n! yk]
=[xn
n! yk] {ykD(x)k
k!
}∞0
=[xn
n!
] {D(x)k
k!
}∞0
The Exponential Formula
Exponential Generating Functions
Solving the question
The number of ways to build a graph with nvertices and k connected components?is just
h(n, k) = H(x, y)[xn
n! yk]
= eyD(x)[xn
n! yk]
=[xn
n! yk] {ykD(x)k
k!
}∞0
=[xn
n!
] {D(x)k
k!
}∞0
The Exponential Formula
Exponential Generating Functions
Solving the question
The number of ways to build a graph with nvertices and k connected components?is just
h(n, k) = H(x, y)[xn
n! yk]
= eyD(x)[xn
n! yk]
=[xn
n! yk] {ykD(x)k
k!
}∞0
=[xn
n!
] {D(x)k
k!
}∞0
The Exponential Formula
Exponential Generating Functions
Solving the question
The number of ways to build a graph with nvertices and k connected components?is just
h(n, k) = H(x, y)[xn
n! yk]
= eyD(x)[xn
n! yk]
=[xn
n! yk] {ykD(x)k
k!
}∞0
=[xn
n!
] {D(x)k
k!
}∞0
The Exponential Formula
Exponential Generating Functions
Thanks For Watching!
Bibliography:
Generatingfunctionology by Herbert S. Wilfhttps://www.math.upenn.edu/∼wilf/gfologyLinked2.pdf
The Exponential Formula
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