Bijections on two variations of noncrossing partitions

A (set) partition of [n] = {1, 2, . . . , n} is a collection of mutually disjoint nonempty subsets, called blocks, of [n] whose union is [n]. We will write a partition as a sequence of blocks (B1, B2, . . . , Bk) such that min(B1) < min(B2) < · · · < min(Bk). There are two natural representations of a partition. Let π = (B1, B2, . . . , Bk) be a partition of [n]. The partition diagram of π is the simple graph with vertex set V = [n] and edge set E, where (i, j) ∈ E if and only if i and j are in the same block which does not have an integer between them. For example, see Figure 1. The canonical word of π is the word a1a2 · · · an, where ai = j if i ∈ Bj . For instance, the canonical word of the partition in Figure 1 is 123124412. For a word τ , a partition is called τ-avoiding if its canonical word does not contain a subword which is order-isomorphic to τ . A partition is noncrossing if the edges of its partition diagram do not intersect. It is easy to see that a partition is noncrossing if and only if it is 1212-avoiding. Let π be a partition and let k be a nonnegative integer. A k-distant crossing of π is a set of two edges (i1, j1) and (i2, j2) of the partition diagram of π satisfying i1 < i2 ≤ j1 < j2 and j1 − i2 ≥ k. A partition π is called k-distant noncrossing if π has no k-distant crossings. Note that 1-distant noncrossing partitions are just noncrossing partitions. Our main objects are 2-distant noncrossing partitions and 12312-avoiding partitions. Let NC2(n) denote the set of 2-distant noncrossing partitions of [n]. Let P12312(n) denote the set of 12312-avoiding partitions of [n].