{"paper":{"title":"Fixed points and cycles of parking functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Martin Rubey, Mei Yin","submitted_at":"2024-03-25T18:47:48Z","abstract_excerpt":"A parking function of length $n$ is a sequence $\\pi=(\\pi_1,\\dots, \\pi_n)$ of positive integers such that if $\\lambda_1\\leq\\cdots\\leq \\lambda_n$ is the increasing rearrangement of $\\pi_1,\\dots,\\pi_n$, then $\\lambda_i\\leq i$ for $1\\leq i\\leq n$. The index $i$ is a fixed point of the parking function $\\pi$ if $\\pi_i=i$. More generally, for $m\\geq 1$, the indices $(i_1, \\dots, i_m)$ where the $i_j$'s are all distinct constitute an $m$-cycle of the parking function $\\pi$ if $\\pi_{i_1}=i_2, \\pi_{i_2}=i_3, \\dots, \\pi_{i_{m-1}}=i_m, \\pi_{i_m}=i_1$. In this paper we obtain some exact results on the num"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.17110","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.17110/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}