Catalan numbers and noncommutative Hilbert schemes

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.