The distribution on permutations induced by a random parking function

Abstract

A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure on $S_n$. We initiate a study of some properties of this distribution. Let $P_n^{\text{park}}$ denote this distribution on $S_n$ and let $P_n$ denote the uniform distribution on $S_n$. In particular, we obtain an explicit formula for $P_n^{\text{park}}(\sigma)$ for all $\sigma\in S_n$. Then we show that for all but an asymptotically $P_n$-negligible set of permutations, one has $P_n^{\text{park}}(\sigma)\in\left(\frac{(2-\epsilon)^n}{(n+1)^{n-1}},\frac{(2+\epsilon)^n}{(n+1)^{n-1}}\right)$. However, this accounts for only an exponentially small part of the $P_n^{\text{park}}$-probability. We also obtain an explicit formula for $P_n^{\text{park}}(\sigma^{-1}_{n-j+1}=i_1,\sigma^{-1}_{n-j+2}=i_2,\cdots, \sigma^{-1}_n=i_j)$, the probability that the last $j$ cars park in positions $i_1,\cdots, i_j$ respectively, and show that the $j$-dimensional random vector $(n+1-\sigma^{-1}_{n-j+l}, n+1-\sigma^{-1}_{n-j+2},\cdots, n+1-\sigma^{-1}_{n})$ under $P_n^{\text{park}}$ converges in distribution to a random vector $(\sum_{r=1}^jX_r,\sum_{r=2}^j X_r,\cdots, X_{j-1}+X_j,X_j)$, where $\{X_r\}_{r=1}^j$ are IID with the Borel distribution. We then show that in fact for $j_n=o(n^\frac16)$, the final $j_n$ cars will park in increasing order with probability approaching 1 as $n\to\infty$. We also obtain an explicit formula for the expected value of the left-to-right maximum statistic $X_n^{\text{LR-max}}$, which counts the total number of left-to-right maxima in a permutation, and show that $E_n^{\text{park}}X_n^{\text{LR-max}}$ grows approximately on the order $n^\frac12$.