Noncrossing Partitions in Surprising Locations

Abstract

1. Introduction.Certain mathematical structures make a habit of reoccuring in the most diverselist of settings. Some obvious examples exhibiting this intrusive type of behaviorinclude the Fibonacci numbers, the Catalan numbers, the quaternions, and themodular group. In this article, the focus is on a lesser known example: the non-crossing partition lattice. The focus of the article is a gentle introduction to thelattice itself in three of its many guises: as a way to encode parking functions, asa key part of the foundations of noncommutative probability, and as a buildingblock for a contractible space acted on by a braid group. Since this article is aimedprimarily at nonspecialists, each area is briefly introduced along the way.The noncrossing partition lattice is a relative newcomer to the mathematicalworld. First defined and studied by Germain Kreweras in 1972 [33], it caught theimagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29],[37], [39], [40], [45], and has come to be regarded as one of the standard objectsin the field. In recent years it has also played a role in areas as diverse as low-dimensional topology and geometric group theory [9], [12], [13], [31], [32] as wellas the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49],[50]. Due no doubt to its recent vintage, it is less well-known to the mathematicalcommunity at large than perhaps it deserves to be, but hopefully this short paperwill help to remedy this state of affairs.2. A motivating example.Before launching into a discussion of the noncrossing partition lattice itself, wequickly consider a motivating example: the Catalan numbers. The Catalan num-bers are a favorite pastime of many amateur (and professional) mathematicians. Inaddition, they also have a connection with the noncrossing partition lattice (Theo-rem 3.1).Example 2.1(Catalan numbers). The Catalan numbers are the numbers C