{"paper":{"title":"On a balanced property of derangements","license":"","headline":"","cross_cats":["cs.NA","math.CV"],"primary_cat":"math.NA","authors_text":"Miklos Bona","submitted_at":"2006-06-12T13:02:22Z","abstract_excerpt":"We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of cycles of a randomly selected derangement of length $n$, then the probability that $k$ is congruent to a given $r$ modulo a given $q$ converges to $1/q$. Finally, we generalize our results to $a$-derangements, which are permutations in which each cycle is longer than $a$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606277","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/math/0606277/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"}