Properties of Peaks in Parking Functions

In 2013, Billey, Burdzy, and Sagan proved that there are 2n−1 permutations of length n with no peaks. In this paper, we discuss generalizations of their results where instead of permutations, we investigate parking functions with no peaks. In particular, we study certain subsets of parking functions and enumerate peaks by analyzing their valleys and plateaus. We also analyze the bijection between nondecreasing parking functions with repeating digits and certain labeled Dyck paths. 

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This research was done in collaboration with Summar Ellis, Dr. Pamela E. Harris, Morgan Hobson, Dr. Marissa Loving, Jospeh Rennie, and Dr. Gordon Rojas Kirby as part of the Mathematical Sciences Research Institute Undergraduate Program.