On the Noncrossing Partitions of a Cycle

This article defines the paritions of a finite set structured in a cycle which possesses the property that a pair of points belonging to a class and a pair of points belonging to another class cannot be in a crossed way. It establishes that these partitions form a lattice and it specifies some of the descriptive and enumerative properties of the lattice; it computes in particular the Möbius function. 1. Definitions In all that follows, we call cycle the pair (M, c) formed by (1) a nonempty, finite set M of cardinality m, (2) a circular bijection c of M into itself, where the word circular means that for all x ∈ M and for all i ∈ {1, 2, . . . ,m − 1} we have c(x) 6= x. The elements of M are called points. Let A be any nonempty subset of M , and let x ∈ A. If kx is the least positive integer such that cx(x) ∈ A, we put cx(x) = d(x). It is clear that d(x) defines a circular bijection of A into itself; (A, d) is thus a cycle, and we call it the trace of (M, c) over A. For all pairs (x, y) of distinct points of M , we call δ(x, y) (distance from x to y) the least positive integer k such that c(x) = y; thus we have, for all pairs {x, y}, δ(x, y) + δ(y, x) = m. Given two disjoint pairs {x, y} and {u, v}, we say that the pairs are crossed if the integer δ(x, y) is between the lesser and greater of the the two integers δ(x, u) and δ(x, v), else the two are uncrossed. Any two disjoint subsets A and B of M are said to be noncrossing if there does not exist two crossed pairs contained in A and B respectively; in particular if at least one of the two disjoint subsets A and B is a singleton (subset of cardinality 1), A and B are necessarily noncrossing. In certain cases we will consider two noncrossing subsets A and B of M which possess the following property: there exists two points x and y such that x ∈ A, y ∈ B, c(x) ∈ B, c(y) ∈ A. If this is so, we say that the two subsets A and B are adjacent. We note that one of the two adjacent subsets can be a singleton {x}; the other then contains c(x) and c(x). Given a cycle (M, c), we call noncrossing partition of M a partition in which any two distinct classes are noncrossing. Date: March 7, 2005. Translated by Berton A. Earnshaw. 1