Reviews

Solomon, L. (1967), "The Burnside algebra of a finite group", J. Comb. Theory, 1: 603–615

## **Carmichael 1937**

*Every possible algebra A[s] is an algebra A[pn] defined as in $102, by means of a doubly transitive group of prime-power degree pn and order pn(pn-1), while, conversely, such a doubly transitive group is induced by the totality of transformations of the form (2) on the marks of such an algebra.*

Carmichael, Groups of finite order, 1937, page 401

There is a deep identity between

1) **doubly transitive groups**, in which **two points** determine the entire permutation
2) **algebras** (which are here not necessarily commutative fields), that have **O and 1**
3) **geometrical lines**, determined by **two points** and getting coordinates by field numbers.
The Carmichael connection is absolutely natural, given two subgroups - the translations and the rotations in a doubly transitive group.

Using the species notations, we have **F" = X.X....X. Stick two, stuck all**.

### **the distributivity**

From the definitions of addition and multiplication we have the following propositions:

u
ρ

u
σ

=

u
γ

{\displaystyle u\_\rho + u\_\sigma = u\_\gamma }

    if

h
ρ

h
σ

=

h
γ

{\displaystyle h\_\rho h\_\sigma = h\_\gamma }

u
i

×

u
ρ

=

u
λ

{\displaystyle u\_i \times u\_\rho = u\_\lambda}

    if

m
ρ

m
i

=

m
λ

{\displaystyle m\_\rho m\_i = m\_\lambda }

u
i

×

u
σ

=

u
ν

{\displaystyle u\_i \times u\_\sigma = u\_\nu}

    if

m
σ

m
i

=

m
ν

{\displaystyle m\_\sigma m\_i = m\_\nu}

u
i

×

u
τ

=

u
α

{\displaystyle u\_i \times u\_\tau= u\_\alpha}

    if

m
τ

m
i

=

m
α

{\displaystyle m\_\tau m\_i = m\_\alpha}

u
λ

u
ν

=

u
β

{\displaystyle u\_\lambda + u\_\nu= u\_\beta}

    if

h
λ

h
ν

=

h
β

{\displaystyle h\_\lambda h\_\nu = h\_\beta}

In order to establish the required relation it is necessary and sufficient to show that

# α
β

{\displaystyle \alpha = \beta}

h
β

=

h
λ

h
ν

=

m
λ

−
1

h
1

m
λ

⋅

m
ν

−
1

h
1

m
ν

{\displaystyle h\_\beta = h\_\lambda h\_\nu = m\_\lambda^{-1} h\_1 m\_\lambda \cdot m\_\nu ^{-1} h\_1 m\_\nu }

=
(

m
ρ

m
i

)

−
1

h
1

(

m
ρ

m
i

)
⋅
(

m
σ

m
i

)

−
1

h
1

(

m
σ

m
i

)

{\displaystyle = ( m\_\rho m\_i )^{-1} h\_1 ( m\_\rho m\_i ) \cdot ( m\_\sigma m\_i )^ {-1} h\_1 ( m\_\sigma m\_i ) }

=

m
i

−
1

h
ρ

h
σ

m
i

=

m
i

−
1

h
γ

m
i

{\displaystyle = m\_i^{-1} h\_\rho h\_\sigma m\_i = m\_i^{-1} h\_\gamma m\_i }

=

m
i

−
1

m
γ

−
1

h
1

m
γ

m
i

=
(

m
γ

m
i

)

−
1

h
1

(

m
γ

m
i

)

{\displaystyle = m\_i^{-1} m\_\gamma^{-1} h\_1 m\_\gamma m\_i = ( m\_\gamma m\_i )^{-1} h\_1 ( m\_\gamma m\_i ) }

=

m
α

−
1

h
1

m
α

=

h
α

{\displaystyle =m\_\alpha^{-1} h\_1 m\_\alpha = h\_\alpha }

## 1989: The one-one correspondence between molecules and permutation groups

The relations in the title reveals to be a ono-to-one correspondence.

```
J. Labelle, Y.N. Yeh
*The relation between burnside rings and combinatorial species*
Journal of Combinatorial Theory, Series A Volume 50, Issue 2, March 1989, Pages 269-284
https://www.sciencedirect.com/science/article/pii/0097316589900198

```

Let H be the stabilizer of one element of the S\_n set,

Let

S

λ

⊂

S

n

{\displaystyle \mathbf S\_\lambda \subset \mathbf S\_n }

 be the set of permutations of type

# λ
(

λ
1

,

λ
2

,
…
)

{\displaystyle \lambda = (\lambda\_1, \lambda\_2, \dots) }

F
i
x
(
σ
# )

{
τ
H

σ
τ
# H
τ
H

}

τ
∈

S

n

{\displaystyle Fix(\sigma) = | \{ \tau H | \sigma \tau H = \tau H \} \_ {\tau \in \mathbf S\_n} | }

Each

τ
H

{\displaystyle \tau H }

 may be written in exactly

H

{\displaystyle |H|}

 modes using different tau's

F
i
x
(
σ
)
.

H

=

{
τ
∈

S

n

σ
τ
# H
τ
H
}

=

{
τ
∈

S

n

τ

−
1

σ
τ
∈

H
}

=
#
{
τ
∈

S

n

τ

−
1

σ
# τ

h
1

∨

τ

−
1

σ
# τ

h
2

∨
…
}

{\displaystyle = \# \{\tau \in \mathbf S\_n | \tau^{-1} \sigma \tau = h\_1 \or \tau^{-1} \sigma \tau = h\_2 \or \dots \}}

=

∑

h
∈
H

{
τ
∈

S

n

τ

−
1

σ
# τ
h
}

{\displaystyle = \sum\_{h \in H } \{ \tau \in \mathbf S\_n | \tau^{-1} \sigma \tau = h \} }

 and, since h and σ are the same type:

=

∑

h
∈
H
∩

S

λ

{
τ
∈

S

n

τ

−
1

σ
# τ
h
}

{\displaystyle = \sum\_{h \in H\cap \mathbf S\_\lambda } \{ \tau \in \mathbf S\_n| \tau^{-1} \sigma \tau = h \} }

=
A
u
t
(
σ
)
.

H
∩

S

λ

=
n
!
.

S

λ

−
1

.

H
∩

S

λ

Finally we have

F
i
x
(
σ
)

d
e
g
r
e
e

=

H
∩

S

λ

S

λ

Then

Z

S
p
e
c
i
e
s

=

1

n
!

∑

σ
∈

S

n

F
i
x
(
σ
)

x
σ

=

1

d
e
g
r
e
e
.

H

∑

σ
∈

S

n

F
i
x
(
σ
)

x
σ

{\displaystyle Z\_{Species} = {1 \over n!} \sum\_{\sigma \in \mathbf S\_n} Fix(\sigma) x^\sigma

= {1 \over degree.|H| } \sum\_{\sigma \in \mathbf S\_n} Fix(\sigma) x^\sigma }

=

1

H

∑

σ
∈

S

n

H
∩

S

λ

S

λ

x
σ

=

1

H

∑
λ

H
∩

S

λ

x
σ

=

Z

S
t
a
b

## **1993**

### **The equations DkY = Xn** in combinatorial species

this one was reference for my A000001

```
Dayanand S.Rajan
Discrete Mathematics Volume 118, Issues 1–3, 1 August 1993, Pages 197-206
https://www.sciencedirect.com/science/article/pii/0012365X9390061W

```

### **Burnside marks reloaded**

Burnside marks reloaded. Table of marks for S1, S2, S3, S4, S5.

```
Dayanand Rajan
Journal of Combinatorial Theory, Series A Volume 62, Issue 1, January 1993, Pages 93-106
https://www.sciencedirect.com/science/article/pii/009731659390073H

```

## **1995**: Square root of X

−
1
−
X
−

E
2

(
X
)

X
2

X

E
2

(
X
)
−

X
3

−

E
2

(

E
2

(
X
)
)

E
2

(
X

)
2

−
2

X
2

E
2

(
X
)

E
2

(

X
2

)
.
.
.

{\displaystyle -1-X-E\_2(X)+X^2+XE\_2(X)-X^3-E\_2(E\_2(X))
+E\_2(X)^2 -2X^2E\_2(X)+E\_2(X^2)+ ... }

```
Pierre Bouchard, Yves Chiricota, Gilbert Labelle
*Arbres, arborescences et racines carrées symétriques*
Discrete Mathematics Volume 139, Issues 1–3, 24 May 1995, Pages 49-56
https://www.sciencedirect.com/science/article/pii/0012365X94001242

```

## **2000**: binomial coefficients for F(1+X)

Partially labeled species,

F
(
1
X
# )

∑
N

(

F
G

)

N
(
X
)

{\displaystyle F(1+X) = \sum\_N {F \choose G} N(X)}

```
Pierre Auger,Gilbert Labelle, Pierre Leroux
*Generalized Binomial Coefficients for Molecular Species*
Journal of Combinatorial Theory, Series A, Volume 91, Issues 1–2, July 2000, Pages 15-48
https://www.sciencedirect.com/science/article/pii/S0097316500930887

```

## 2002 F(X + Y +···+ Z)

```
*Pierre Auger, Gilbert Labelle, and Pierre Leroux*
Combinatorial Addition Formulas and Applications
Advances in Applied Mathematics Volume 28, Issues 3–4, April 2002, Pages 302-342
https://www.sciencedirect.com/science/article/pii/S0196885801907766

```

## 2013

Maria Gillespie
A006963 with n nodes:

# R
X
.
C
(
T
)

{\displaystyle R=X.C(T)}

Maria Gillespie, Counting the labeled plane trees on *n* vertices

## 2014 Hugo

```
HugoTremblay Gilbert Labelle Srečko Brleka Alexandre Blondin Massé
*Exhaustive generation of atomic combinatorial differential operators*
Theoretical Computer Science Volume 536, 29 May 2014, Pages 62-69
https://www.sciencedirect.com/science/article/pii/S0304397514001455

```

?

Naughton

Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group

## to check

The Subgroups of M24, or
How to Compute the Table of Marks
of a Finite Group
Gotz Pfeiffer

## References

* André Joyal, *Une théorie combinatoire des séries formelles*, Advances in Mathematics 42:1–82 (1981).
* Labelle, Jacques. *Quelques espèces sur les ensembles de petite cardinalité.*, Ann. Sc. Math. Québec 9.1 (1985): 31-58.
* J. Labelle and Y.N. Yeh, *The Relation Between Burnside Rings and Combinatorial Species*, Journal of Combinatorial Theory Series A 50, (1989) 269–284
* Yves Chiricota, *Classification des espèces moléculaires de degré 6 et 7*, Ann. Sci. Math. Québec 17 (1993), no. 1, 1 l–37.
* François Bergeron, Gilbert Labelle, Pierre Leroux, *Théorie des espèces et combinatoire des structures arborescentes*, LaCIM, Montréal (1994). English version: *Combinatorial Species and Tree-like Structures*, Cambridge University Press (1998).
* Federico G. Lastaria, An invitation to Combinatorial Species. (2002)

Marcelo Fiore, Tom Leinster Objects of Categories as Complex Numbers https://arxiv.org/abs/math/0212377 on seven trees in one.

## Note

* Federico G. Lastaria, An invitation to Combinatorial Species.
* Brent A. Yorgey, Species and Functors and Types, Oh My!
* Brent A. Yorgey, Species and Functors and Types, Oh My! video
* John C. Baez, Toby Bartels, Being an Octopus, 2004

Pentru tabele :

* Jaques Labelle,  *Quelques espèces sur les ensembles de petite cardinalité*, Ann. Sci. Math. Québec 9 (1985), no. 1, 31-58.
* Yves Chiricota, *Classification des espèces moléculaires de degré 6 et 7*, Ann. Sci. Math. Québec 17 (1993), no. 1, 11-37.
* "On Transitive Permutation Groups" by J.H. Conway, A. Hulpke, and J. McKay, LMS J. Comput. Math. 1 (1998), 1-8.
* Nota?iile Maple ale grupurilor tranzitive de grad = 15

* http://web.mac.com/xgviennot/Xavier\_Viennot/cours\_files/Ch3.pdf 1988
* Newton Institute, 2008
* http://algo.inria.fr/flajolet/Publications/books.html

* André Joyal, „Une théorie combinatoire des séries formelles”, *Advances in Mathematics* 42:1-82 (1981).
* François Bergeron, Gilbert Labelle, Pierre Leroux, *Théorie des espèces et combinatoire des structures arborescentes*, LaCIM, Montréal (1994). English version: *Combinatorial Species and Tree-like Structures*, Cambridge University Press (1998).