Partition of A Set - Noncrossing Partitions

Noncrossing Partitions

A partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing provided that for any two 'cells' C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all the elements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cell called C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.