Atomic and molecular species

[Christian G.Bower]

Goetz Pfeiffer contacted me with information from the Labelle/Yeh
paper on the sequence A005227.


> They describe it as
>
>   the number of (isomorphism classes of) atomic species of degree $n$
>   which are not non-trivial substitution.
>

This is essentially the same as what I said:

> 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.

So my argument that A005227 should be 111 instead of 107:

There are 130 atomic species of order 8 according to A005226.  If 107
involve only trivial substitution (i.e. of the form X(A) or A(X) where
X is the species of singletons) then there are 23 built from
nontrivial substitution.  They must be of the form 2*4, 4*2 or
2*2*2. (8*1 or 1*8 leads to trivial substitution.)  Since I have a
list of all species of orders 2 and 4, this should be straightforward.

2*2*2

E2(E2(E2))
E2(E2(X^2))

2*4

E2(E4)
E2(E4+-)
E2(E2(E2)) was listed in 2*2*2
E2(X*E3)
E2(E2^2)
E2(P4bic)
E2(C4)
E2(X*C3)
E2(X^2*E2)
E2(E2(X^2)) was listed in 2*2*2
E2(X^4)

4*2

E4(E2)
E4(X^2)
E4+-(E2)
E4+-(X^2)
E2(E2)(E2) same as E2(E2(E2)) from 2*2*2
E2(E2)(X^2) same as E2(E2(X^2)) from 2*2*2
P4bic(E2)
P4bic(X^2)
C4(E2)
C4(X^2)
E2(X^2)(E2) same as E2(E2^2) from 2*4
E2(X^2)(X^2) same as E2(X^4) from 2*4

giving 19 unique case as opposed to the predicted 23, hence there must
be 111 not formed this way rather than the listed 107.  Of course a
better way would be to go through all the species of order 8, but that
isn't happening any time soon.  Therefore, I will submit the changes.

Christian