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).