{"id":"W1542393439","doi":"10.37236/579","title":"Exterior Pairs and Up Step Statistics on Dyck Paths","year":2011,"lang":"en","type":"article","venue":"The Electronic Journal of Combinatorics","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"Toronto Metropolitan University","funders":"National Science Council","keywords":"Equidistributed sequence; Mathematics; Combinatorics; Statistic; Automorphism; Bijection, injection and surjection; Discrete mathematics; Statistics; Bijection","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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001054021,0.0004435452,0.001051964,0.002165572,0.001276304,0.001812293,0.0009980196,0.0005493538,0.002427465],"category_scores_gemma":[0.005880444,0.0003839248,0.0005453188,0.001112818,0.003365157,0.003014251,0.00297532,0.001740982,0.0002730351],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001017439,"about_ca_system_score_gemma":0.000504757,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000605209,"about_ca_topic_score_gemma":0.0005231485,"domain_scores_codex":[0.9985384,0.0002681777,0.00007418879,0.0003618158,0.000435437,0.0003220761],"domain_scores_gemma":[0.9939728,0.002710375,0.001233616,0.0005561414,0.0005235775,0.001003418],"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":[0.0002318091,0.00006483257,0.009635899,0.00005684991,0.00002153577,0.0002241484,0.0005430386,0.005467583,0.007593952,0.9646088,0.0006193655,0.01093213],"study_design_scores_gemma":[0.00006306112,0.0002519034,0.01468099,0.0000398528,0.00004762401,0.0009944062,0.0007011393,0.06967858,0.01372222,0.8946893,0.005008114,0.0001227351],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9272785,0.000206685,0.06138964,0.0001391846,0.00004679009,0.0000162807,0.0001602549,0.0001288242,0.01063388],"genre_scores_gemma":[0.9910816,0.0001206958,0.006475381,0.00004785436,0.00004368065,0.00004241919,0.0001968014,0.0000425885,0.001948899],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.002427465,"threshold_uncertainty_score":0.008120656,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03760349564316957,"score_gpt":0.2830632771789675,"score_spread":0.2454597815357979,"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."}}