Intervals of Antichains and Their Decompositions
  An antichain of subsets is a set of subsets such that no subset in the antichain is a proper subset of any other subset in the antichain. The Dedekind number counts the total number of antichains of subsets of an n-element set. This paper investigates the interval structure of the lattice of antichains. Several partitioning theorems and counting formulas for the size of intervals are derived. 
Patrick De Causmaecker, Stefan De Wannemacker, Jay Yellen
math.CO