Enumeration of (k, 2)-noncrossing partitions

A partition Π of the set [n] = {1, 2, . . . , n} is a collection B1, B2, . . . , Bd of nonempty disjoint subsets of [n]. The elements of a partition are called blocks. We assume that B1, B2, . . . , Bd are listed in the increasing order of their minimum elements, that is minB1 < minB2 < · · · < minBd. The set of all partitions of [n] with d blocks is denoted by P (n, d). The cardinality of P (n, d) is the well-known Stirling number of the second kind [8], which is usually denoted by S(n, k). Any partition Π can be written in the canonical sequential form π1π2 · · ·πn, where i ∈ Bπi (see, e.g. [4]). From now on, we identify each partition with its canonical sequential form. For example, if Π = {1, 4}, {2, 5, 7}, {3}, {6} is a partition of [7], then its canonical sequential form is π = 1231242 and in such a case we write Π = π.