Atomic and molecular species

[p.j.cameron at qmul.ac.uk]

The expert on enumerating permutation groups is Alexander Hulpke at
Colorado State. He can give you guaranteed correct figures for
conjugaacy classes of subgroups of S_n up to around 10 or 11.

Peter Cameron.

P.S. Wreath product of permutation groups is standard terminology.

On Mon, Feb 20, 2006 at 02:54:20PM -0800, Christian G.Bower wrote:
> 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
> 
> 
>