{"id":"W7117425606","doi":"10.61091/ars165-07","title":"Counting permutations in \\(S_{2n}\\) and \\(S_{2n+1}\\)","year":2025,"lang":"","type":"article","venue":"Ars Combinatoria","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"","keywords":"Disjoint sets; Conjecture; Constructive; Square (algebra); Constructive proof; Symmetric group","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0007158421,0.0006183361,0.0004571562,0.001179867,0.001344062,0.001923857,0.0009577272,0.0009226002,0.007585945],"category_scores_gemma":[0.00322135,0.000275537,0.0004719167,0.00158213,0.00216696,0.002991284,0.001096549,0.0007810575,0.001172944],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001011955,"about_ca_system_score_gemma":0.0008433569,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001171408,"about_ca_topic_score_gemma":0.00193641,"domain_scores_codex":[0.9989941,0.0001793813,0.00007344164,0.0003154972,0.0002249297,0.000212572],"domain_scores_gemma":[0.9979885,0.0008347117,0.0003881736,0.0003764981,0.0002423492,0.0001697839],"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":[0.000273296,0.00008487926,0.008121516,0.0001597663,0.00002385364,0.000425228,0.0006934227,0.005144053,0.01175858,0.8680169,0.009185299,0.09611332],"study_design_scores_gemma":[0.00003358302,0.00009826547,0.004694648,0.00003969375,0.00003229302,0.0008486429,0.0002566751,0.03298701,0.01894891,0.9197059,0.02229622,0.00005822009],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.7036567,0.0003017859,0.2093187,0.001429851,0.0002785748,0.0002188703,0.001405053,0.0007273306,0.08266312],"genre_scores_gemma":[0.9355415,0.0002353438,0.04516077,0.0003033383,0.0001669234,0.0002406307,0.001267868,0.000200732,0.01688296],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.007585945,"threshold_uncertainty_score":0.02537751,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01697249890892259,"score_gpt":0.3070683590954668,"score_spread":0.2900958601865442,"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."}}