Comments on A104525
Thomas Wieder (thomas.wieder(AT)t-online.de), May 31 2008

A104525 gives the number of hierarchical orderings among the parts of the 
integer partitions of the integer n. In order to explain this structure we first
consider the ordered partitions of partitions (opops) of the integer n, that is 
sequence A055887(n).  Let the colon ":" be a position separator. We think of 
hierarchies  where ":" separates two consecutive ranks. The semicolon ";" just 
separates two opops. For n=3 we have A055887(3)=8.  These 8 opops are:

{{1},{1},{1}}; {1,1,1}; {1}:{1}:{1}; {1,1}:{1}; {1}:{{1},{1}}; {{1},{1}}:{1}; 
{1}:{1,1}; {1},{1,1}.

An equivalent (and may be more appealing) representation is:

1,1,1; 3; 1:1:1; 2:1; 1:{1,1}; {1,1}:1; 1:2; 1,2.

We see that an opop can be interpreted as an hierarchical ordering. In an 
hierarchical ordering either unlabeled or labeled elements are distributed among 
labeled ranks. The nature of the elements in the case of an opop is particular, 
the elements are subsets of the n-set {1,1,...,1} composed of n unlabeled 
elements "1".

In the second step, we introduce a second separator "|". Two parts (subsets) of
a partition belong to two different opops if they are separated by "|". In this
way we introduce multiple opopos for a single partition.
For our example with n=3 we get the 12 structures

{{1},{1},{1}}; {1},{1},{1}; {1,1,1}; {1}:{1}:{1}; {1},{1}:{1}; {1,1}:{1}; 
{1}:{{1},{1}}; {1},{{1},{1}}; {{1},{1}}:{1}; {1}:{1,1}; {1},{1,1}; {{1},{1,1}}.

Or equivalently:

1,1,1; 1|1|1; 3; 1:1:1;	1|1:1; 2:1; 1:{1,1}; 1|{1,1}; {1,1}:1; 1:2; 1|2; 1,2.

Thus the hierarchical ordering A104525(n) follows from the application of the 
two separators ":" and "|" on the parts on the integer partition of n.  A104525 
is therefore the partition analogon to sequence A034691.  Whereas for A104525 
the elements are parts ("groups"), for A034691  the elements may be thought as 
unlabeled atoms ("individuals").  The labeled analoga for this point of view are
A109186 and A075729. 



The e.g.f. of A104525 follows from an Euler transform of the opop A055887. If 
a(n) denotes A055887(n), then (in Maple notation)

e.g.f. = mul(1/(1-x^n)^a(n),n=1..inf)

For definitions of transforms on integer sequences see:
M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, 
Linear Algebra and its Applications, 226-228 (1995), 57-72.


We can use combstruct to actually construct the structures A104525(n).
The combstruct command is

Test := [V,{V=Set(U),U=Sequence(S,card>=1),S=Set(T,card>=1),T=Set(Z,card>=1)},
unlabelled]; 
seq(count(Test,size=j),j=1..20);

For n=3 the comstruct command "allstructs(Test,size=3);" returns 

[Set(Sequence(Set(Set(Z), Set(Z, Z)))),
Set(Sequence(Set(Set(Z)), Set(Set(Z)), Set(Set(Z)))),
Set(Sequence(Set(Set(Z), Set(Z)), Set(Set(Z)))),
Set(%1, Sequence(Set(Set(Z, Z)))),
Set(Sequence(Set(Set(Z), Set(Z), Set(Z)))), Set(%1, %1, %1),
Set(Sequence(Set(Set(Z, Z)), Set(Set(Z)))),
Set(%1, Sequence(Set(Set(Z), Set(Z)))),
Set(%1, Sequence(Set(Set(Z)), Set(Set(Z)))),
Set(Sequence(Set(Set(Z)), Set(Set(Z), Set(Z)))),
Set(Sequence(Set(Set(Z, Z, Z)))),
Set(Sequence(Set(Set(Z)), Set(Set(Z, Z))))]
%1 := Sequence(Set(Set(Z))).