A Hopf algebra of parking functions

TL;DR

This paper introduces a Hopf algebra structure on parking functions by connecting free cumulants, symmetric functions, and permutation representations, revealing new algebraic insights into parking functions.

Contribution

It constructs a Hopf algebra of parking functions based on their relation to symmetric functions and permutation representations, providing a new algebraic framework.

Findings

The sequence of symmetric functions $(f_n)$ corresponds to the Frobenius characteristic of permutation representations.

A Hopf algebra structure on parking functions is explicitly constructed and analyzed.

The work links free cumulants, symmetric functions, and parking functions in a novel way.

Abstract

If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions $(f_n)$. We prove that $(f_n)$ is the Frobenius characteristic of the natural permutation representation of $\SG_n$ on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail.