{"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":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001499328,0.000301312,0.0005435496,0.0001204551,0.0001761302,0.00003875749,0.0006173838,0.0001098292,0.0000385207],"category_scores_gemma":[0.0009824191,0.0002061289,0.0001079802,0.0002793614,0.0001325816,0.0001603571,0.0001039338,0.0009806488,0.00001158705],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0003095909,"about_ca_system_score_gemma":0.0002623016,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00000228222,"about_ca_topic_score_gemma":0.000002507016,"domain_scores_codex":[0.9975381,0.0002180256,0.00079273,0.0001641327,0.0006397289,0.0006472689],"domain_scores_gemma":[0.996844,0.001191118,0.0009563413,0.000513473,0.0003489832,0.0001460956],"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":[0.0001419696,0.0002923404,0.00004296709,0.00005720941,0.0001042267,0.00002363206,0.001366312,6.82562e-7,0.00005086969,0.9952039,0.001232837,0.001483055],"study_design_scores_gemma":[0.001337666,0.001916713,0.0000777803,0.00009023378,0.0001635297,0.0002272819,0.0004525275,0.00009273236,0.0009737827,0.9931565,0.001288687,0.0002225855],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.6663122,0.002499261,0.321176,0.0003276628,0.00448915,0.0009054682,0.00001480161,0.0001306468,0.004144804],"genre_scores_gemma":[0.995694,0.0006592778,0.002884262,0.00005782386,0.0001132298,0.000006522533,3.520859e-7,0.00007227562,0.0005122197],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.3293818,"threshold_uncertainty_score":0.8405691,"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."}}