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NONCROSSING PARTITIONS IN SURPRISING LOCATIONS JON MCCAMMOND
 

Summary: NONCROSSING PARTITIONS IN SURPRISING LOCATIONS
JON MCCAMMOND
1. Introduction.
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 non-
crossing 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

  

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

 

Collections: Mathematics