{"id":"W2551475233","doi":"10.46298/dmtcs.2503","title":"Negative $q$-Stirling numbers","year":2015,"lang":"en","type":"article","venue":"Discrete Mathematics & Theoretical Computer Science","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":2,"is_retracted":false,"has_abstract":true,"ca_institutions":"Toronto Metropolitan University","funders":"National Security Agency; Princeton University; Simons Foundation","keywords":"Mathematics; Stirling numbers of the first kind; Stirling number; Partially ordered set; Combinatorics; Binomial coefficient; Stirling numbers of the second kind; Binomial (polynomial); Bijection; Statistics","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":["metaepi_narrow","sts"],"consensus_categories":[],"category_scores_codex":[0.003127724,0.0005092324,0.0007316327,0.0002254678,0.000420712,0.000397549,0.001847818,0.0001305374,0.00008890246],"category_scores_gemma":[0.005270408,0.0004100214,0.000174155,0.001276549,0.003472276,0.0006481849,0.001080123,0.0004425526,0.0002157451],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0002332217,"about_ca_system_score_gemma":0.0002627587,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000002445526,"about_ca_topic_score_gemma":9.932644e-7,"domain_scores_codex":[0.9952395,0.0001276301,0.0009059626,0.0007923243,0.001913112,0.001021448],"domain_scores_gemma":[0.9947134,0.001992346,0.0003816047,0.00136241,0.0007142924,0.0008359038],"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":[0.00001345164,0.0001871535,0.000007270606,0.00008645405,0.00001890103,0.00001641962,0.00601385,0.00005829444,0.0001324118,0.9928452,0.0002521792,0.0003684012],"study_design_scores_gemma":[0.0005827844,0.0001711453,0.000001805602,0.0001680856,0.00004578417,0.00005721658,0.0009259229,0.1360357,0.002312144,0.8591888,0.00004429254,0.0004663666],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"methods","genre_scores_codex":[0.04536046,0.00001271728,0.9165928,0.000306967,0.001107755,0.0006187671,0.000006477695,0.0004730886,0.03552099],"genre_scores_gemma":[0.4869244,0.000001125606,0.512701,0.00007723916,0.0001813169,0.00002172837,8.779018e-7,0.00004975871,0.00004256376],"genre_candidate":"methods","genre_consensus":"methods","teacher_disagreement_score":0.441564,"threshold_uncertainty_score":0.9998352,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03782610600975795,"score_gpt":0.3254781697518571,"score_spread":0.2876520637420992,"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."}}