{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:31:21Z","timestamp":1762191081805,"version":"build-2065373602"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this article, we establish new results on the probabilistic parking model (introduced by Durmi\u0107, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p \\in [0,1]$. For any $m \\leq n$ and $p \\in [0,1]$, we study the parking preference of the last car, denoted $a_m$, and determine the conditional distribution of $a_m$ and compute its expected value. We show that both formulas depict explicit dependence on the probability parameter $p$. We study the case where $m=cn$ for some $0&lt;c&lt;1$ and investigate the asymptotic behavior and show that the presence of ``extra spots'' on the street significantly affects the rate at which the conditional distribution of $a_m$ converges to the uniform distribution on $[n]$. Even for small $\\varepsilon=1-c$, an $\\varepsilon$-proportion of extra spots reduces the convergence rate from $1\/\\sqrt{n}$ to $1\/n$ when $p\\neq 1\/2$. Additionally, we examine how the convergence rate depends on $c$, while keeping $n$ and $p$ fixed. We establish that as $c$ approaches zero, the total variation distance between the conditional distribution of $a_m$ and the uniform distribution on $[n]$ decreases at least linearly in $c$.<\/jats:p>","DOI":"10.37236\/13864","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:13:37Z","timestamp":1762190017000},"source":"Crossref","is-referenced-by-count":0,"title":["Probabilistic (m, n)-Parking Functions"],"prefix":"10.37236","volume":"32","author":[{"given":"Pamela","family":"Harris","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodrigo","family":"Ribeiro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mei","family":"Yin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,11,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/13864\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/13864\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T17:13:37Z","timestamp":1762190017000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/13864"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,3]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,10,3]]}},"URL":"https:\/\/doi.org\/10.37236\/13864","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,3]]},"article-number":"P4.39"}}