Metered Parking Functions and Prescribed Lucky Cars

We enumerate metered parking functions by their lucky cars, allowing arbitrary predetermined departure schedules. The fibers of the parking outcome map yield a uniform affine formula valid for every street size and a positive expansion in packed outcomes. For a prescribed lucky set, we prove eventual polynomiality, determine the degree and leading coefficient, and obtain a graphical bound on component spans. For a common meter this bound is attained and depends explicitly on the positions of the unlucky cars. Summing the leading coefficients gives the chromatic polynomial of the overlap graph; the positive expansion gives nonnegative Newton coefficients. We also give an exact algorithm that is polynomial at fixed occupancy width. For meter one, a reduced rational generating function and a positive recurrence count every number of cars, including more cars than sites, while an interval-partition formula determines the exact polynomial onset for every prescribed lucky set. Finally, a finite boundary comparison identifies the highest coefficients of the total counting polynomial, including the next coefficient that detects chronology.