Descent refinement of the peak–tie recurrence

Refine the finite local-state recurrence for parking-function peak and tie statistics by incorporating descents while retaining a finite local state.

Background

The paper develops a recurrence for the joint peak-and-tie polynomial of parking functions by inserting copies of a new maximal letter and tracking local insertion types. This recurrence supplies a recursive answer to previously posed questions about the joint distribution of peaks and ties.

The authors identify a further unresolved refinement: augment the recurrence with descent information without losing the finite local-state structure that makes the insertion method tractable.