Rational Noncrossing Coxeter-Catalan Combinatorics

TL;DR

This paper introduces rational noncrossing objects for all finite Coxeter groups, proves they are counted by rational Coxeter-Catalan numbers, and uses advanced algebraic methods to establish these results.

Contribution

It provides a new combinatorial framework for rational noncrossing objects and a uniform proof of their enumeration using Hecke algebra and Fourier transform techniques.

Findings

Defined rational noncrossing objects for any finite Coxeter group

Proved these objects are counted by rational Coxeter-Catalan numbers

Extended results to rational noncrossing parking objects

Abstract

We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.