Closed-form enumeration of outcome permutations O_{m,n}(I) for fixed lucky set I with more spots than cars

Determine a closed-form formula for |O_{m,n}(I)|, where O_{m,n}(I) is the set of outcomes in S_{m,n} (permutations of [m] with n−m placeholders for empty spots) that arise from (m,n)-parking functions whose lucky cars are exactly the subset I ⊆ [m] (with 1 ∈ I).

Background

The authors generalize their results to (m,n)-parking functions with m ≤ n, defining outcomes as permutations in S_{m,n} where X denotes an empty spot. They characterize the outcomes associated with a fixed lucky set I and count outcomes in certain structured cases.

As in the n-spot case, however, they note that a general closed-form enumeration of |O_{m,n}(I)| for arbitrary I remains open.