{"paper":{"title":"Group Theory of the Kolakoski Sequence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Noah MacAulay","submitted_at":"2026-05-14T00:59:30Z","abstract_excerpt":"Run-length decoding is an operation on sequences in which a positive integer $a$ is replaced by a run(sequence of repeated elements) of length $a$. Iterated run-length decodings applied to sequences with alphabets consisting of pairs of positive integers $\\{p, q\\}$ have attracted attention from mathematicians, most notably in their role defining the well-known Kolakoski sequence. $n$-th-iterated run-length decodings are controlled by naturally associated permutation automata $A^{p,q}_n$. Here we study the transformation groups $\\mathcal{K}^{p,q}_n$ of these automata. They are subgroups of the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"48099d468852d75b6a64ec764a2ae261bbcf0bb7e2090a4ae1b757153d2fd589"},"source":{"id":"2605.14234","kind":"arxiv","version":1},"verdict":{"id":"1a7e53dd-7b6c-44bf-98ea-0b77216a631b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:42:53.011340Z","strongest_claim":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.","one_line_summary":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch.","pith_extraction_headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths."},"references":{"count":5,"sample":[{"doi":"","year":1939,"title":"Rufus Oldenburger, Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466","work_id":"a1f06b72-18fb-4db4-abd1-725721e76050","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"Kolakoski, Self Generating Runs, Problem 5304, American Math","work_id":"3eed7634-77da-48a4-8e07-b974219bbb35","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"The Kolakoski sequence and related conjectures about orbits","work_id":"41b9176e-3671-42df-90e0-85a25207cc96","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"Notes on the Kolakoski sequence","work_id":"9d6b0f83-56db-4a8d-9832-3f7cae865664","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Branch groups","work_id":"f2549d1b-fc81-4e1b-bbe5-c705324e8e20","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":5,"snapshot_sha256":"2c2c97cfeb31176f2595baaf543ac4ff0ebd272b29e09ba28878b712df8ac6c2","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"088287ea65fe4512288c80f224a5cd7b372f5ec3dcab37f5031272fc48610359"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}