On Some Discrete Statistics of Parking Functions. Appears in the Journal of Integer Sequences.

Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon.

A parking function is a tuple whose nondecreasing rearrangement satisfies the condition that no element exceeds its index. We study parking functions by their ascents, descents, and tie. By using multiset Eulerian polynomials, we give a generating function for the number of parking functions of length n with i descents. We present a recursive formula for the number of parking functions of length n with descents at a specified set of locations. We establish a bijection between the set of parking functions with a certain descent set S, and the set of parking functions where the descents are a reversal of S. As a special case, we enumerate the parking functions of length n with descents only in the first i positions. We prove this using a connection to standard young tableaux, whose enumeration is known. We also study peaks and valleys of parking functions. We show that the set of parking functions with no peaks and no ties is enumerated by the Catalan numbers, and the set of parking functions with no valleys and no ties is enumerated by the Fine numbers. We conclude our study by characterizing when a parking function is uniquely determined by its statistic encoding; a word indicating which indices in the parking function are ascents, descents, and ties. We provide open problems throughout. 

Defective Parking Functions and Young Tableaux. Pre-print available on the Arxiv.

Rebecca E. Garcia, Pamela E. Harris, Alex Moon, Aaron Ortiz, Lauren J. Quesada, Cynthia Marie Rivera Sánchez, and Dwight Anderson Williams II. 

An (m,n,d)-defective parking function is a parking function where m cars attempt to park along a street with n parking spots, but d cars fail to park. We show that (m,n,d)-defective parking functions are invariant under permuting action of the symmetric group. We also find a way to compute the defect of an arbitrary tuple. We enumerate orbits of (m,n,d)-defective parking functions by connecting them to certain Young tableaux, whose enumeration is known. Using a multinomial formula for the size of an orbit, we conclude with a new formula for the number of (m,n,d)-defective parking functions.