Given a partial order of elements, one can use topological sorting to produce a sorted list of elements. For example, if we have the partial order A->B and A->C, then the possible topological sort results are [A,B,C] and [A,C,B]. 


I am interested in producing a sorted list of **sets** [$S\_1, \ldots, S\_k$] that satisfy the partial order. (The sets $S\_i$ partition the elements.) Here, the requirements are: 


1. for $i = 1 \ldots k-1$, $\exists e\_1 \in S\_i,e\_2 \in S\_{i+1}$ s.t. $e\_1 < e\_2$
2. for each set $S\_i$, $\nexists e\_1, e\_2 \in S\_i$ such that $e\_1 < e\_2$ or $e\_2 < e\_1$


In our example, the only correct sorted list of sets is [{A},{B,C}]. Given a partial order, how many possible sorted lists of sets exist? Is there a name for this kind of sorting? Any pointers are appreciated. 



Topological sort of partial order into sorted sets
EDIT: This answer was for a previous version of the question.


There is usually no such list: consider the case where some element is incomparable to everything else.


