Counting Stirling permutations with a fixed set of lucky cars

Develop general results that count Stirling permutations which, when viewed as parking functions, have a prescribed fixed set of lucky cars; that is, determine formulas for the number of Stirling permutations for which exactly the cars in a given subset are lucky.

Background

In reviewing related work on Stirling permutations viewed as parking functions, the authors summarize results that handle particular cases, including counts when only the first car is lucky and when the number of lucky cars is maximal.

They note explicitly that general results for counting Stirling permutations with a fixed set of lucky cars are still unknown.