A Shifted Parking Function Symmetric Function

TL;DR

This paper introduces a shifted analogue of the parking function symmetric function, providing explicit expansions in shifted symmetric function bases and discussing properties of potential shifted parking function analogues.

Contribution

It defines a shifted parking function symmetric function and describes its explicit expansions, advancing the understanding of shifted symmetric functions related to parking functions.

Findings

Explicit expansion of $ ext{SH}_n$ in shifted symmetric bases

Discussion of properties for a potential shifted parking function analogue

No current shifted analogue for parking functions itself

Abstract

We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a shifted analogue for parking functions themselves, but some desirable properties of such an analogue are discussed.