Multichains, non-crossing partitions and trees

In a previous paper El], we proved results -about the enumer;ation of certain types of chains in the non-crossing partition lattice T, and its, generalizations. In this paper we present bijections to certain classes of trees which reprove one theorem [l, Corollary 3.41 and provide a combinatoridi proof for the other [I, Theorem 5.31. We begin with a review of the definitions. A set partition X = {B,, BZ, . . . , Z&} of the <set {1,2, . . . , m}= [m] is called non-crossing (n.c.) if there do not exist four numbers a < b Cc <d such that a, c E Bi and b, d E Bi and if j. Let T, be the set of all n.c. partitions of [m] ordered by refinement. That is, XS Y if each block of X is contained in a block of Y. T, is a lattice and was first studied by Kreweras [5] and Poupard [S]. Define a R.C. 2-partition of the set [m] to be: a set