{"paper":{"title":"On Deranged Unit-Interval Parking Functions and the Deranged Bell Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yahia Djemmada","submitted_at":"2026-06-30T21:17:45Z","abstract_excerpt":"Unit-interval parking functions of length $n$ are enumerated by the Fubini numbers $F_n$ and are in explicit bijection with the ordered set partitions of $[n]$. We use this bijection to single out the unit-interval parking functions whose associated ordered set partition is \\emph{deranged} in the sense of Belbachir, Djemmada, and N\\'emeth -- no block occupies the position indexed by its minimum element -- and call them the \\emph{deranged unit-interval parking functions} $\\DUPF_n$. Since the bijection restricts to the deranged objects, $|\\DUPF_n| = \\tilde F_n$, the $n$-th deranged Bell number. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01273","kind":"arxiv","version":1},"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/2607.01273/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"}