k-partitions, matroids and linear spaces

[Gordon Royle]

Hi SeqFans..

I am trying to find information about the following things...

First, consider a partition of a number n.. we can also think of this  
as a collection of subsets of the set {1,2.., n} such that every  
element is in exactly one subset, and two collections are isomorphic  
if there is a permutation in Sym(n) taking one to the other.

Second, consider a 2-partition of a set of size n - this is a  
collection of subsets of {1,2, ..., n} such that every PAIR of  
elements is in exactly one subset, with isomorphism defined  
naturally. This is also known as a pairwise balanced design of index  
1 as far as I can work out.  These are also equivalent to linear  
spaces which are enumerated in A056642, where it is, correctly,  
pointed out that this number is 1 greater than the number of simple  
rank 3 matroids.

However, what is not EXPLICITLY pointed out is that A056642 is not  
counting isomorphism classes, but rather "labelled" objects. To me,  
this is rather dangerous in that if one says " ... the number of  
graphs that .. " or " .. the number of matroids that .. " then 99% of  
the time, it is isomorphism classes that are being referred to.

So what about isomorphism classes of these things.. well they are  
there in A001200 but now they are called "linear geometries" and now  
they ARE counting isomorphism classes, but fail to mention it (hmm,  
surely we should insist that ONE of the labelled/unlabelled versions  
of a sequence actually identify itself as being labelled or  
unlabelled.. i suggest for safety that BOTH should have explicit  
mention of it). Also it would be nice if there were references from  
the labelled version to the unlabelled and vice versa..

  And what about the isomorphism classes of simple rank 3 matroids  
(of which there are exactly 1 fewer than (isomorphism classes of )  
linear spaces? Well these are present as well, in A058731, and now  
labelled fully as "Number of isomorphism classes .." but somehow the  
author has forgotten that the number is one less than the linear  
geometries, because A001200 has entries up to 12 elements, while  
A058731 (which is just 1 less than A001200) only has entries up to 8  
elements.

[Those wondering where this "one fewer" keeps cropping up is that  
linear spaces count an n-element line as a valid linear space, but it  
is not a rank 3 matroid for the good reason that it has rank 2 when  
viewed as a matroid]


Now my question is whether the numbers of 3-partitions are anywhere  
in OEIS? (i.e. a collection of subsets such that every TRIPLE of  
elements occurs in a unique subset..). They are not there under that  
name as far as I can tell, but with the plethora of different names  
maybe there is another name for them that I just don't know?


Basically they are quite natural.. just a 3-design of index 1, but  
with any-old block sizes.

Thanks

Gordon