Non-commutative Frobenius characteristic of generalized parking functions -- Application to enumeration

TL;DR

This paper introduces a recursive framework for generalized parking functions, computes their non-commutative characteristic, and derives enumeration formulas, providing new insights into their algebraic and combinatorial structures.

Contribution

It presents a novel recursive definition of generalized parking functions as a species and computes their non-commutative characteristic, linking algebraic structures to enumeration.

Findings

Derived enumeration formulas for generalized parking functions

Provided interpretations in various bases of non-commutative symmetric functions

Analyzed an inclusion-exclusion formula by Kung and Yan

Abstract

We give a recursive definition of generalized parking function that allows us to view them as a species. From there we compute a non-commutative characteristic of the generalized parking function module, and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non-commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.