{"paper":{"title":"The counts of parity-odd vs even late-growing permutations in the group $\\mathbb{S}_{2p-1}$ differ by $(-1)^p$","arxiv_id":"2610.35779","primary_cat":"math.CO","submitted_at":"2026-10-26T17:18:17+00:00","version_compiled":"v1"},"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 v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"A permutation $\\sigma\\in\\mathbb{S}_n$ of objects $\\{1,\\ldots,n\\}$ is called \\emph{late-growing} if its partial sums never exceed the respective partial sums of the `average' permutation (with the arithmetic mean $n^{-1}\\cdot\\sum_{i=1}^n i = (n+1)/2$ of $1,\\ldots,n$ on average at each entry $\\sigma(i)$): $\\sum_{i=1}^k \\sigma(i) \\leqslant k\\cdot (n+1)/2$ for all positive $k\\leqslant n$.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.12,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.79,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"result","text":"We prove that for $n=2p-1$ odd, the set $\\Phi_p\\subseteq \\mathbb{S}_{2p-1}$ of late-growing permutations is partitioned into parity-even vs odd elements, $\\Phi_p = \\Phi_p^{+} \\sqcup \\Phi_p^{-}$, in an almost symmetric way: $| \\Phi_p^{+} | - | \\Phi_p^{-} | = (-1)^{p+1}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.94,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.98,"id":"s2","own":true,"role":"method","text":"The proof is constructive: by viewing the permutations $\\sigma\\in\\Phi_p$ as constrained walks on the integral lattice $\\mathbb{Z}$, we describe two filters that pair all but one element in $\\Phi_p^{+}$ versus $\\Phi_p^{-}$; the remaining walk $\\sigma^*$ with monotonically growing step amplitudes determines the dominant parity subset.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.28,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"We prove that for $n=2p-1$ odd, the set $\\Phi_p\\subseteq \\mathbb{S}_{2p-1}$ of late-growing permutations is partitioned into parity-even vs odd elements, $\\Phi_p = \\Phi_p^{+} \\sqcup \\Phi_p^{-}$, in an almost symmetric way: $| \\Phi_p^{+} | - | \\Phi_p^{-} | = (-1)^{p+1}$.","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":3534,"cost_usd_reported":0.0},"compiled_at":"2026-10-27","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.03,"states_limitation":0.13,"human_participants":0.01,"controlled_comparison":0.08,"randomized_assignment":0.01,"states_quantitative_result":0.97,"compares_with_prior_results":0.05,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.84,"text":"We prove that for $n=2p-1$ odd, the set $\\Phi_p\\subseteq \\mathbb{S}_{2p-1}$ of late-growing permutations is partitioned into parity-even vs odd elements, $\\Phi_p = \\Phi_p^{+} \\sqcup \\Phi_p^{-}$, in an almost symmetric way: $| \\Phi_p^{+} | - | \\Phi_p^{-} | = (-1)^{p+1}$.","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 v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":3,"sentences_in_abstract":3},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}