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Noncrossing Partitions in Surprising Locations

Summary: Noncrossing Partitions in
Surprising Locations
Jon McCammond
1. INTRODUCTION. Certain mathematical structures make a habit of turning up in
the most diverse 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 low-dimensional 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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics