Noncrossing Partitions in Surprising Locations

Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing partition lattice. The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space acted on by a braid group. Since this article is aimed primarily at nonspecialists, each area is briefly introduced along the way. The noncrossing partition lattice is a relative newcomer to the mathematical world. First defined and studied by Germain Kreweras in 1972 [33], it caught the imagination 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 objects in the field. In recent years it has also played a role in areas as diverse as lowdimensional topology and geometric group theory [9], [12], [13], [31], [32] as well as 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 mathematical community at large than perhaps it deserves to be, but hopefully this short paper will help to remedy this state of affairs.