Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes

TL;DR

This paper establishes a combinatorial bijection linking zonotope lattice points, parking functions, and Dyck paths, motivated by noncommutative Hilbert schemes, and constructs a semi-orthogonal decomposition of their derived category.

Contribution

It introduces an explicit $S_n$-equivariant bijection connecting zonotope points, parking functions, and Dyck paths, and applies tilting bundles to decompose the derived category of noncommutative Hilbert schemes.

Findings

Bijection between zonotope points and parking functions

Restriction to regular orbits and Dyck paths counted by Fuss-Catalan numbers

Construction of semi-orthogonal decomposition for derived categories

Abstract

We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of $m$-parking functions of length $n$. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m,n)$-Dyck paths, the number of which is given by the Fuss-Catalan number $A_{n}(m,1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.