Closed-form enumeration of outcome permutations O_n(I) for fixed lucky set I

Determine a closed-form formula for |O_n(I)|, where O_n(I) is the set of permutations in S_n that arise as parking outcomes of length-n parking functions whose lucky cars are exactly the subset I ⊆ [n] (with 1 ∈ I), i.e., the cars that park in their preferred spots are precisely those in I.

Background

The paper introduces O_n(I) as the set of outcome permutations for length-n parking functions whose lucky set (cars that park in their preferred spot) is exactly I. The authors fully characterize which permutations occur as outcomes for a given lucky set and derive counts in certain special cases, such as when the first k cars are lucky.

Despite this characterization, obtaining a general closed-form formula for |O_n(I)| remains elusive; the authors explicitly state that the overall enumeration problem, beyond special configurations, is still open.