Formulas

(

∂

∂

x
1

,

∂

∂

x
2

,

∂

∂

x
2

,
⋯

)

P
H

(

z
1

,
2

z
2

,
3

z
3

,
⋯
)

{\displaystyle P\_G \left( \frac {\partial}{\partial x\_1}, \frac {\partial}{\partial x\_2}, \frac {\partial}{\partial x\_2}, \cdots \right) P\_H ( z\_1, 2z\_2, 3z\_3, \cdots) }

Z
F

(

∂

∂

x
1

,
2

∂

∂

x
2

,
3

∂

∂

x
2

,
⋯

)

Z
G

(

x
1

,

x
2

,

x
3

,
⋯
)

{\displaystyle Z\_F \left( \frac {\partial}{\partial x\_1}, 2\frac {\partial}{\partial x\_2}, 3\frac {\partial}{\partial x\_2}, \cdots \right)
Z\_G ( x\_1, x\_2, x\_3, \cdots) }

## BLL- Main Book

### page 17, the cycle index

a)

Z
F

(

x
1

,

x
2

,

x
3

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.
# )

∑

n
≥
0

1

n
!

(

∑

σ
∈

S
n

f
i
x
F
[
σ
]

x
1

σ
1

x
2

σ
2

x
3

σ
3

…

)

{\displaystyle Z\_F ( \ x\_1, x\_2, x\_3 ,...) = \sum\_{n \ge 0} {1 \over n!} \left ( \sum\_{\sigma \in S\_n} fix F[\sigma] x\_1^{\sigma\_1}x\_2^{\sigma\_2} x\_3^{\sigma\_3} \dots \right)}

### page 18: exponential and type series from cycle index

a)

F
(
x
# )

Z
F

(

x
,
0
,
0
,
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.
.
)

{\displaystyle F(x) = Z\_F ( \ x, 0, 0,...) }

b)

F
~

(
x
# )

Z
F

(

x

,

x
2

,

x
3

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.
.
)

{\displaystyle \tilde F(x) = Z\_F ( \ x \ , x^2, x^3...) }

### page 29...43 Operations

page 29: Sum

(
F
G
)
[
U

# U
U

F
[
U
]
G
[
U
]

{\displaystyle (F+G)[U] = \sum\_{U=U} F[U]+G[U] }

page 32: Product

(
F
.
G
)
[
U
# ]

∑

U
1

U
2

=
U

F
[

U
1

]
×
G
[

U
2

]

{\displaystyle (F.G)[U] = \sum\_{U\_1+U\_2=U} F[U\_1] \times G[U\_2] }

page 38: Composition

(
F
(
G
)
)
[
U
# ]

∑

U
1

U
2

U
3

# ⋯
U

F
[
π
]
×
G
[

U
1

]
×
G
[

U
2

]
×
G
[

U
3

]
×
…

{\displaystyle (F(G))[U] = \sum\_{U\_1+U\_2+U\_3+ \dots=U} F[\pi] \times G[U\_1] \times G[U\_2] \times G[U\_3] \times \dots}

page 43: Derivative

F
′

[
U
# ]

∑

U
′

⋆

F
[

U
′

]

{\displaystyle F'[U] = \sum\_{U'+ \star} F[U'] }

### page 39, composition series

a)

(
F
∘
G
)
(
x
# )
F
(
G
(
x
)
)

{\displaystyle (F \circ G )(x) = F(G(x))\ }

b)

F
∘
G

~

=

Z
F

(

G
~

(
x
)
,

G
~

(

x
2

)
,

G
~

(

x
3

)
,
.
.
.

)

{\displaystyle \tilde {F \circ G} = Z\_F \left ( \ \tilde G(x), \ \tilde G(x^2), \ \tilde G(x^3), ... \right ) }

c)

Z

F
∘
G

(

x
1

,

x
2

,

x
3

,
.
.
.
# )

Z
F

(

Z
G

(

x
1

,

x
2

,
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.
.
)
,

Z
G

(

x
2

,

x
4

,
.
.
.
)
,
.
.
.
)

{\displaystyle Z\_{ F \circ G}(x\_1, x\_2, x\_3,... ) = Z\_F ( \ Z\_G(x\_1, x\_2,... ), \ Z\_G(x\_2, x\_4,... ),... )\ }

### **page 90 - multisort**

.

a)

F
(
G
,
H
)
(
x
,
y
# )
F
(
G
(
x
,
y
)
,
H
(
x
,
y
)

{\displaystyle F(G,H) (x,y) = F(G(x,y), H(x,y) }

b)

F
(
G
,
H
)

~

(
x
,
y
# )

Z
F

(

/

G
~

(
x
,
y
)
)
,

G
~

(

x
2

,

y
2

)
,
.
.
.
;

H
~

(
x
,
y
)
,

H
~

(

x
2

,

y
2

)
,
.
.
.
)

{\displaystyle \tilde {F(G,H)} (x,y) = Z\_F ( / \tilde G(x,y)), \tilde G(x^2, y^2),...; \tilde H(x,y), \tilde H(x^2, y^2),...) }

c)

Z

F
(
G
,
H
)

=

Z
F

(

Z
G

,

Z
H

# )

Z
F

(

Z

G
,
1

,

Z

G
,
2

,
.
.
.
;

Z

H
,
1

,

Z

H
,
2

,
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.
.
)

{\displaystyle Z\_{F(G,H)} = Z\_F(Z\_G,Z\_H) = Z\_F( Z\_{G,1}, Z\_{G,2},...; Z\_{H,1}, Z\_{H,2},... ) }

where

Z

G
,
k

(

x
1

,

x
2

,

x
3

,
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.
.
# )

Z
G

(

x
k

,

x

2
k

,

x

3
k

,
.
.
.
)

{\displaystyle Z\_{G,k} (x\_1, x\_2, x\_3,... ) = Z\_G (x\_k, x\_{2k},x\_{3k}, ...) }

### **The Wreath Product**

In group theory the composition of species coresponds to the wreath product.

For example, the table of **Cyc3(Ens2)** is :

(
F
∘
G
)
[
A
# ]

∑

π
∈
P
[
A
]

(
F
[
π
]
×

Π

B
∈
π

G
[
B
]
)
.

{\displaystyle (F \circ G)[A] = \sum\_{\pi \in P[A]} (F[\pi] \times \Pi\_{B \in \pi} G[B]).}