{"paper":{"title":"Introduction to Partially Ordered Patterns","arxiv_id":"math/0603122","primary_cat":"math.CO","submitted_at":"2006-03-05T12:10:21+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v2","sentences":[{"i":0,"p":0.91,"id":"s0","own":false,"role":"background","text":"We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.13,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.91,"limitation":0.0,"future_work":0.0,"contribution":0.09,"interpretation":0.0}},{"i":1,"p":0.68,"id":"s1","own":true,"role":"contribution","text":"We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.61,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.32,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.68,"interpretation":0.0}},{"i":2,"p":0.86,"id":"s2","own":true,"role":"contribution","text":"Moreover, we discuss q-analogue for a result on non-overlapping segmented POPs.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.79,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.14,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.86,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"future_work","text":"Finally, we suggest several open problems for further research.","hedge":{"p":0.63,"label":"hedged_possibility"},"own_p":0.77,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":1.0,"contribution":0.0,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution.","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":3681,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.03,"states_limitation":0.12,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.32,"compares_with_prior_results":0.45,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":0.86,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.14,"proof_or_derivation":0.86,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":0.79,"value":"theoretical result","probabilities":{"review":0.21,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.79,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v2"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":4,"sentences_in_abstract":4},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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