On Some Discrete Statistics of Parking Functions

A parking function is a sequence $\alpha=(a_1,a_2,\ldots,a_n)\in[n]^n$ whose nondecreasing rearrangement $\beta=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq i$ for all $1\leq i\leq n$ . We study parking functions by their ascents (indices at which $a_i<a_{i+1}$ ), descents (indices at which $a_i>a_{i+1}$ ), and ties (indices at which $a_i=a_{i+1}$ ). By using multiset Eulerian polynomials, we give a generating function for the number of parking functions of length $n$

with

descents. We present a recursive formula for the number of parking functions of length $n$

with descents at a specified subset of $[n-1]$

. We establish the set of parking functions with descent set $I$

and the set of parking functions with descent set $J=\{n-i : i\in I\}$ are in bijection, and hence these sets have the same cardinality. As a special case, we show that the number of parking functions of length $n$

with descents at the first $k$

indices is given by $\frac{1}{n}\binom{n}{k}\binom{2n-k}{n-k-1}$ . We prove this by bijecting to the set of standard Young tableaux of shape $((n-k)^2,1^k)$

, which are enumerated by the same formula. We also study peaks and valleys of parking functions, which are indices at which $a_{i-1}<a_i>a_{i+1}$ and $a_{i-1}>a_i<a_{i+1}$ , respectively. 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.