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We consider the set of length-$k$ scattered factors of a given word w, called here $k$-spectrum and denoted $\\ScatFact_k(w)$. We prove a series of properties of the sets $\\ScatFact_k(w)$ for binary strictly balanced and, respectively, $c$-balanced words $w$, i.e., words over a two-letter alphabet where the number of occurrences of each "},"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":"1904.09125","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.FL","submitted_at":"2019-04-19T09:09:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d9404a5bc23a725be199ea020bd7b9f1da02b92b65b0db223274b0bbd4fc3c3b","abstract_canon_sha256":"7e0ff6b04c833cca25a6af671d4002f42fb3b89972df089c3ee31f1f86b57629"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:14.643190Z","signature_b64":"VbqxFcF4p8LUxjXDGt06DJTMAlKoSDyRWNXGeojDdaL3RIL3Fwu4M1XO1+rL/Znaj2a6ptzA3cHnN2PVSaEeCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d12f66079bace80d27f9f67eec8505870441f7d2fb4cb1dba4e780718315477","last_reissued_at":"2026-05-17T23:45:14.642688Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:14.642688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"k-Spectra of weakly-c-Balanced Words","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.FL","authors_text":"Dirk Nowotka, Florin Manea, Joel D. Day, Pamela Fleischmann","submitted_at":"2019-04-19T09:09:49Z","abstract_excerpt":"A word $u$ is a scattered factor of $w$ if $u$ can be obtained from $w$ by deleting some of its letters. That is, there exist the (potentially empty) words $u_1,u_2,..., u_n$, and $v_0,v_1,..,v_n$ such that $u = u_1u_2...u_n$ and $w = v_0u_1v_1u_2v_2...u_nv_n$. We consider the set of length-$k$ scattered factors of a given word w, called here $k$-spectrum and denoted $\\ScatFact_k(w)$. 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