{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T23:34:22Z","timestamp":1773444862768,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $X_n^{\\text{LR-max}}$, the statistic that counts the total number of lleft-to-right maxima in a permutation, and show that $E_n^{\\text{park}}X_n^{\\text{LR-max}}$ grows approximately on the order $n^\\frac12$. In terms of car parking, the left-to-right maximum statistic counts the number of cars that park in the first available space.<\/jats:p>","DOI":"10.37236\/13842","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T22:22:03Z","timestamp":1773440523000},"source":"Crossref","is-referenced-by-count":0,"title":["The Distribution on Permutations Induced by a Random Parking Function"],"prefix":"10.37236","volume":"33","author":[{"given":"Ross","family":"Pinsky","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,3,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p50\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T22:22:11Z","timestamp":1773440531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13842","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,13]]},"article-number":"P1.50"}}