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:

$$u_\rho + u_\sigma = u_\gamma $$   if    $$ h_\rho h_\sigma = h_\gamma $$

$$u_i \times u_\rho = u_\lambda$$   if    $$ m_\rho m_i = m_\lambda $$

$$u_i \times u_\sigma = u_\nu$$   if    $$ m_\sigma m_i = m_\nu$$

$$u_i \times u_\tau= u_\alpha$$   if    $$ m_\tau m_i = m_\alpha$$

$$u_\lambda + u_\nu= u_\beta$$   if    $$ h_\lambda h_\nu = h_\beta$$

In order to establish the required relation it is necessary and sufficient to show that $$\alpha = \beta $$

$$h_\beta = h_\lambda h_\nu = m_\lambda^{-1} h_1 m_\lambda  \cdot   m_\nu ^{-1} h_1 m_\nu $$

$$ = ( m_\rho m_i )^{-1} h_1 ( m_\rho m_i ) \cdot  ( m_\sigma m_i )^ {-1} h_1  ( m_\sigma m_i )  $$

$$ = m_i^{-1} h_\rho h_\sigma m_i =  m_i^{-1} h_\gamma m_i   $$

$$ = m_i^{-1} m_\gamma^{-1} h_1 m_\gamma m_i  = ( m_\gamma m_i )^{-1} h_1  ( m_\gamma m_i )  $$

$$ =m_\alpha^{-1} h_1 m_\alpha = h_\alpha $$

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 $$ \mathbf S_\lambda \subset \mathbf S_n $$ be the set of permutations of type $$ \lambda = (\lambda_1, \lambda_2, \dots) $$


 * $$ Fix(\sigma) = | \{ \tau H | \sigma \tau H = \tau H \} _ {\tau \in \mathbf S_n} | $$

Each $$ \tau H $$ may be written in exactly $$| H |$$ modes using different tau's


 * $$ Fix(\sigma).|H| = | \{\tau \in \mathbf S_n | \sigma \tau H = \tau H \}  |  = | \{\tau \in \mathbf S_n | \tau^{-1} \sigma \tau \in \ H \} | $$


 * $$ = \# \{\tau \in \mathbf S_n | \tau^{-1} \sigma \tau = h_1 \or \tau^{-1} \sigma \tau = h_2 \or \dots \}$$


 * $$ = \sum_{h \in H }  \{ \tau \in \mathbf S_n  | \tau^{-1} \sigma \tau = h \}  $$ and, since h and σ are the same type:


 * $$ = \sum_{h \in H\cap \mathbf S_\lambda }  \{ \tau \in \mathbf S_n| \tau^{-1} \sigma \tau = h \}  $$


 * $$ = Aut(\sigma) . | H \cap \mathbf S_\lambda | = n!. |\mathbf S_\lambda |^{-1}.|H \cap \mathbf S_\lambda| $$

Finally we have


 * $$ { Fix(\sigma) \over degree } = { |H \cap \mathbf S_\lambda| \over |\mathbf S_\lambda | } $$

Then


 * $$ 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 \over |H| } \sum_{\sigma \in \mathbf S_n} { |H \cap \mathbf S_\lambda| \over |\mathbf S_\lambda | } x^\sigma

= {1 \over |H| } \sum_\lambda |H \cap \mathbf S_\lambda|  x^\sigma = Z_{Stab} $$

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
$$-1-X-E_2(X)+X^2+XE_2(X)-X^3-E_2(E_2(X)) +E_2(X)^2 -2X^2E_2(X)+E_2(X^2)+ ... $$

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,

$$ F(1+X) = \sum_N {F \choose G} N(X)$$

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: $$R=X.C(T)$$ 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

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).