A Bijection between Two Variations of Noncrossing Partitions

A (set) partition of [n] = {1, 2, . . . , n} is a collection of disjoint 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), for instance, ({1, 4, 8}, {2, 5, 9}, {3}, {6, 7}). Let π = (B1, B2, . . . , Bk) be a partition of [n]. 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 the canonical word of the partition does not contain a subword which is order-isomorphic to τ . A partition is noncrossing if the edges of the diagram of the partition do not intersect. It is easy to see that a partition is noncrossing if and only if it is 1212-avoiding. We will focus on two recent variations of noncrossing partitions. Dan and Kim [1] defined the following. 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 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. By finding a generating function, Dan and Kim [1] showed that the number of 2-distant noncrossing partitions of [n] is equal to the number of Schröder paths of length 2n− 2 with no peaks at even level, which is A007317 in [4]. Using the kernel method, Mansour and Severini [3] proved that this number is equal to the number of 12312-avoiding partitions of [n]. The aim of this note is to find bijections between these objects. One of our ingredients is the bijection of Yan [5] from the set of UH-free Schröder paths of length 2n − 2 to the set of 12312avoiding partitions of [n].