{"id":"W2890957687","doi":"10.1007/s00026-020-00500-9","title":"Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers","year":2020,"lang":"en","type":"article","venue":"Annals of Combinatorics","topic":"Advanced Mathematical Theories and Applications","field":"Physics and Astronomy","cited_by":6,"is_retracted":false,"has_abstract":false,"ca_institutions":"York University","funders":"","keywords":"Catalan number; Binomial coefficient; Fibonacci number; Unimodality; Natural number; Quotient; Sequence (biology); Lucas number; Integer sequence; Integer (computer science)","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":[0.00006630066,0.00008613148,0.0002814156,0.00002651446,0.00003316453,0.000005055608,0.0001325037,0.00002476412,0.00002244873],"category_scores_gemma":[0.00004517364,0.0000836848,0.00007421741,0.0002452516,0.0001677817,0.00005848541,0.00006933296,0.0000633947,0.000001595441],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.000001887758,"about_ca_system_score_gemma":0.00002994843,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00002957154,"about_ca_topic_score_gemma":1.231958e-7,"domain_scores_codex":[0.9993175,0.00001820806,0.0003336167,0.000110329,0.0001206899,0.00009964551],"domain_scores_gemma":[0.9991854,0.0001552613,0.0002464568,0.0001359951,0.0001953189,0.00008159593],"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":[0.00002019361,0.0002289463,0.002065154,0.00006244655,0.00004405115,6.615973e-8,0.0007354094,0.00002905856,0.0004839328,0.9956345,0.0001815691,0.0005146815],"study_design_scores_gemma":[0.0006236034,0.0002263883,0.0003258058,0.00003923349,0.0000378656,1.191897e-7,0.0006850723,0.0005443096,0.04352461,0.9528538,0.001020742,0.000118438],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9528937,0.00007491928,0.03759126,0.0005252876,0.0001477959,0.0002486748,0.0002659289,0.00001456497,0.008237919],"genre_scores_gemma":[0.9996986,0.000008010058,0.0001937511,0.00002071046,0.00002595706,0.000006080782,0.00002964253,0.000008103637,0.000009180744],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.04680491,"threshold_uncertainty_score":0.3412567,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03722687948889762,"score_gpt":0.3140588587923512,"score_spread":0.2768319793034536,"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."}}