A Shifted Parking Function Symmetric Function

We define a “shifted analogue” ${\mathrm{S}\mathrm{H}}_n$ of the parking function symmetric function ${\mathrm{P}\mathrm{F}}_n$. The expansion of ${\mathrm{S}\mathrm{H}}_n$ 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.