Refined enumeration of noncrossing chains and hook formulas

TL;DR

This paper refines the enumeration of noncrossing chains in Coxeter groups, introducing a weighted generalization and an equivalence relation, leading to new formulas including hook length formulas in types A and B.

Contribution

It introduces a one-parameter weighted enumeration of noncrossing chains and an equivalence relation that simplifies their generating functions, extending hook formulas.

Findings

Generalized noncrossing chain enumeration with weights

Established simple generating functions for equivalence classes

Recovered and extended hook length formulas in types A and B

Abstract

In the combinatorics of finite finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this article is to refine the formula. First, we prove a one-parameter generalization, by the considering enumeration of noncrossing chains where we put a weight on some relations. Second, we consider an equivalence relation on noncrossing chains coming from the natural action of the group on set partitions, and we show that each equivalence class has a simple generating function. Using this we recover Postnikov's hook length formula in type A and obtain a variant in type B.