{"id":"W4408567095","doi":"10.61091/jcmcc124-26","title":"Over-inversion numbers: Basic properties, log-concavity and unimodality","year":2025,"lang":"en","type":"article","venue":"Journal of Combinatorial Mathematics and Combinatorial Computing","topic":"Analytic Number Theory Research","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Direction Générale de la Recherche Scientifique et du Développement Technologique","keywords":"Unimodality; Mathematics; Inversion (geology); Combinatorics; Geology; Paleontology","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow"],"consensus_categories":[],"category_scores_codex":[0.004048723,0.0003602591,0.001073683,0.0002758975,0.0004225949,0.0002818271,0.00049027,0.0002498718,0.00001737711],"category_scores_gemma":[0.002303357,0.000301101,0.0001963731,0.0004636478,0.0003099692,0.0002920469,0.000629605,0.0008039815,0.000001856868],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0002271153,"about_ca_system_score_gemma":0.0003482958,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00004369603,"about_ca_topic_score_gemma":0.000001804575,"domain_scores_codex":[0.9967656,0.0003581604,0.001279148,0.0003160285,0.0008318708,0.0004491631],"domain_scores_gemma":[0.9959339,0.001842459,0.0008890317,0.0003965398,0.0006789001,0.000259184],"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":[0.0002421589,0.0006041045,0.001574333,0.0008986961,0.0002499369,0.00002581544,0.0008812776,0.000001331049,0.0007150405,0.9924111,0.001185841,0.001210346],"study_design_scores_gemma":[0.004851364,0.0003325816,0.0001508806,0.0009656324,0.0002369064,0.00004739059,0.0007963098,0.004413001,0.001111211,0.9862087,0.0005999034,0.0002860641],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9889116,0.0003212604,0.002334049,0.0003105328,0.004690127,0.0004046102,0.000002056122,0.00004741849,0.002978331],"genre_scores_gemma":[0.9968576,0.00004853449,0.002322608,0.00004301274,0.0006039088,0.000001763505,5.095167e-7,0.00003387101,0.00008818575],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.007945992,"threshold_uncertainty_score":0.9999441,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0342718434835666,"score_gpt":0.310089536252786,"score_spread":0.2758176927692194,"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."}}