{"id":"W4312121971","doi":"10.48550/arxiv.2212.11556","title":"The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure","year":2022,"lang":"en","type":"preprint","venue":"arXiv (Cornell University)","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":true,"ca_institutions":"Université de Montréal","funders":"","keywords":"Combinatorics; Lattice (music); Mathematics; Order (exchange); Quotient; Discrete mathematics; Physics","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":[0.0006114464,0.0004874762,0.000507955,0.001520724,0.001284265,0.00401711,0.0006410705,0.0006940558,0.005611888],"category_scores_gemma":[0.001432428,0.0003433846,0.0008107569,0.001774107,0.005317837,0.005658411,0.002078389,0.00202724,0.001066063],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001721952,"about_ca_system_score_gemma":0.0008138146,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000788471,"about_ca_topic_score_gemma":0.0006864088,"domain_scores_codex":[0.9991823,0.0001753047,0.0000634879,0.0002146259,0.0002699964,0.00009420417],"domain_scores_gemma":[0.9990607,0.0002831418,0.0001434036,0.0002146571,0.0001583714,0.0001397541],"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":[0.00001281344,0.000007771662,0.0001614057,0.00003059501,0.000003231583,0.00002745562,0.0001110629,0.0002106416,0.000824816,0.9914525,0.0008288722,0.00632888],"study_design_scores_gemma":[0.00000506392,0.00002489093,0.000265776,0.0000198019,0.00000511803,0.0001903011,0.0001554717,0.001972946,0.001236445,0.967025,0.02908593,0.0000131854],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.2407691,0.01391459,0.4475288,0.006659417,0.002308933,0.0001154585,0.0006304733,0.0005405741,0.2875327],"genre_scores_gemma":[0.8074758,0.006461466,0.132661,0.001369813,0.002078907,0.0001862953,0.0006837332,0.0002180646,0.04886492],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.005611888,"threshold_uncertainty_score":0.01877367,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.06677670414517387,"score_gpt":0.2244161983123662,"score_spread":0.1576394941671923,"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."}}