{"id":"W4360958751","doi":"10.1007/s11009-023-09987-0","title":"On Distribution of the Number of Peaks and the Euler Numbers of Permutations","year":2023,"lang":"en","type":"article","venue":"Methodology And Computing In Applied Probability","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":2,"is_retracted":false,"has_abstract":false,"ca_institutions":"Carleton University; University of Manitoba","funders":"Ministry of Science and Technology","keywords":"Mathematics; Sequence (biology); Permutation (music); Distribution (mathematics); Euler's formula; Combinatorics; Descent (aeronautics); Markov chain; Euler number (physics); Discrete mathematics; Backward Euler method; Mathematical analysis; Statistics; Semi-implicit Euler method; Euler equations","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.004180624,0.00008769367,0.0003820371,0.00001729512,0.00006800921,0.000001994936,0.0001330319,0.00008313898,0.000002826124],"category_scores_gemma":[0.004522791,0.00005392977,0.0000465045,0.0003092355,0.0007324985,0.000008393656,0.0001763278,0.0001823747,3.107196e-7],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00001555746,"about_ca_system_score_gemma":0.00001897958,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00001189987,"about_ca_topic_score_gemma":0.000003549123,"domain_scores_codex":[0.9983288,0.0008181695,0.0004676134,0.0001567697,0.0001167681,0.0001118816],"domain_scores_gemma":[0.9840776,0.01523531,0.0003238714,0.0002972178,0.00005422875,0.00001182714],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000101627,0.0000673234,0.00447037,0.0003269494,0.00001639619,5.257591e-8,0.002468579,0.0004110256,0.0001353714,0.9912003,0.000008206634,0.0007937585],"study_design_scores_gemma":[0.0006990416,0.00001215248,0.01880915,0.00004328408,0.00002689404,0.000002175503,0.000315784,0.001474994,0.001233577,0.9773379,0.000002186255,0.00004286229],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9561096,0.000005690273,0.04243818,0.0001213242,0.0001019639,0.0004385749,0.000005063749,0.00001699392,0.0007626432],"genre_scores_gemma":[0.9583688,0.000002106536,0.04158937,0.000006572203,0.000006702252,0.00001445951,0.000001231237,0.000005575946,0.000005209684],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.01441714,"threshold_uncertainty_score":0.5414531,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.09775155175521023,"score_gpt":0.3771160974663824,"score_spread":0.2793645457111721,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}