Atomic and molecular species

[Christian G.Bower]

I want to add comments to A000638, A005226 and A005227 showing their
relationships and have run into a snag.

Short version: Should A005227(8) be 111 instead of the listed 107?

Long version:
 
A000638 Number of permutation groups of degree n; also number of
conjugacy classes of subgroups of symmetric group S_n; also number of
molecular species of degree n.  (Formerly M1244 N0477)
 
 1, 1, 2, 4, 11, 19, 56, 96, 296, 554, 1593, 3094, 10723 (list)  
 
 OFFSET 0,3 ...  EXTENSIONS a(11) corrected and a(12) added by Goetz
Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004
 

  
A005226  Number of atomic species of degree n. 
(Formerly M1563)   +30
2  
 
 0, 1, 1, 2, 6, 6, 27, 20, 130, 124, 598, 640 (list)  
 
 OFFSET  0,4
  
 REFERENCES F. Bergeron, G. Labelle and P. Leroux, Combinatorial
Species and Tree-Like Structures, Camb. 1998, p. 147.

J. Labelle and Y. N. Yeh, The relation between Burnside rings and
combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.
 
Simon Plouffe (plouffe(AT)math.uqam.ca) 
 
 
A005227  Number of atomic species of degree n. 
(Formerly M1029)

 1, 1, 2, 4, 6, 19, 20, 107, 116, 567, 640 (list)  
 
 OFFSET  1,3
  
 REFERENCES J. Labelle and Y. N. Yeh, The relation between Burnside
rings and combinatorial species, J. Combin. Theory, A 50 (1989),
269-284.
 
 CROSSREFS  Cf. A005226. 

------

A combinatorial species is a sum of molecular species that are
themselves not the sum of smaller species.  The molecular species
correspond exactly to permutation groups.

A molecular is the product of atomic species that are not themselves
the product of smaller species.  Species product corresponds to
permutation group direct product.  Thus atomic species correspond to
connected permutation groups where connected means not the direct
product of smaller permutation groups.  A000638 is the Euler transform
of A005226.

What is A005227 which has the same title as A005226?

Here's what I think:

Another species operation is substitution.  An atomic species can be
constructed through substitution operation involving an atomic species
on the left and a molecular species on the right.  I've seen this
described as a wreath product of permutation groups, but I don't know
how widespread that terminology is.  My thought is that this second
"atomic species" sequence is of species that cannot be built up from
sum, product or substitution of smaller species.  If this is true, it
can be calculated from either of the other two sequences and Goetz
Pfeiffer's extention to A000638 can be applied.  But if this is so,
then a(8) should be 111 rather than 107.  The other values are
correct.  (Except for a(11), but it was consistent with the other two
before Goetz' correction to A000638(11) correction.)

I don't have easy access to the Labelle/Yeh paper.  Is anyone familiar
with it to know if that count was an error, or if A005227 is counting
something else?

P.S.  If A005227 is as I believe, then let b(n), c(n) be such that
b(1)=0, b(k)=A005227(k) k>1, c(0)=0, c(k)=A000638(k), k>0.  A005226 is
the Dirichlet convolution of a and b.

Christian