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We use Cartesian lattice paths to characterize abelian squares in binary sequences, and construct a binary word of length $q(q+1)$ avoiding abelian squares of length $\\geq 2\\sqrt{2q(q+1)}$ or greater. We thus prove that the length of the longest binary word avoiding abelian squares of length $2k$ is $\\Theta(k^2)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1012.0524","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-02T18:04:54Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"86adc715162734f583b91df95c04ab21eb5672acea4842488edaa659be713b20","abstract_canon_sha256":"0230d0537575279bb7fb9f38dbb1aeeff2d73c9df9b2bfcce89ccf67263fe646"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:17.732630Z","signature_b64":"Pk1I6jThv2oTnpZY8YQTSVyfyEzu+yMec/0JuhjpyzcWC2lPvKnYd9o+JvXIRu0fV8iaPumtXajDPSC4VUG+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"634da84c87f28ada0990cd2a836387144e49b19a5d6ac39775ad6b3368991f15","last_reissued_at":"2026-05-18T04:34:17.732183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:17.732183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Avoiding Sufficiently Long Abelian Squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Elyot Grant","submitted_at":"2010-12-02T18:04:54Z","abstract_excerpt":"A finite word $w$ is an abelian square if $w = xx^\\prime$ with $x^\\prime$ a permutation of $x$. 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