Number of preferential arrangements for the set partitions of the n-set [1,2,3,...,n].
Comments from Thomas Wieder, May 07 2008

A083355(n) gives the number of orderings onto levels (preferential arrangements) for the set partitions of n. 

Consider an n-set {1,2,3,...,n} and its set partitions. The number of set 
partitions into k subsets is given by the Stirling numbers of the second kind 
S2(n,k). Now we distribute the subsets onto k levels l=1,2,3,...,k. The order 
within a level does not count, thus {1}{2} = {2}{1}. No level may be empty if 
the next level is occupied. We get A083355(n) distributions. 

If we consider a set of n unlabeled elements {1,1,...,1} then we are dealing 
with the integer partitions of n. Thus, we distribute the k parts of an integer 
partition n into k parts onto k levels. The resulting sequence is A055887(n). 

Similar distributions are described by A00079(n) and A000670(n) but for elements
 instead of sets. If we distribute unlabeled elements onto levels, then we 
arrive at A00079(n). For labeled elements, we get A000670(n). In the later case 
one often speaks of preferential arrangements. Another suitable description 
would be to speak of a ranking of elements, where the elements can be labeled or 
unlabeled, atoms or sets etc.

In previous submissions we named such distributions "hierarchical 
orderings", but now we would like to restrict the attribute "hierarchical" for 
those distributions for which the occupation of level l+1 is always smaller (or 
at most equal) to the occupation of level l. From now on, if we consider 
structures like A083355(n) or A055887(n) we will speak of "rankings" or 
(equivalently) "preferential arrangements" or "orderings onto levels". Two 
important rankings for elements are 
A034691(n) and A075729(n). Both structures not only rank their elements but 
additionally split the set of elements into subsets and consequently into 
independent subrankings. 
A034691(n) is related to unlabeled elements and can be found as the Euler 
transform  of powers of 2 = A000079(n). A075729(n) pertains to labeled elements 
and is the Exp transform of A000670(n).