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Given a function f on (n-1) bits, let N(f; r) be the number of functions generating a De Bruijn sequence of order n which are obtained by changing r locations in the truth table of f. We prove a formula for the generating function \\sum_r N(\\ell; r) y^r when \\ell is a linear function.\n  The proof uses a weighted Matrix Tree Theorem and a description of the in-trees (or rooted trees) in the n-bit De Bruijn graph as perturbations of the Hamiltonian paths in the same graph."},"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":"1705.07835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-22T16:29:08Z","cross_cats_sorted":[],"title_canon_sha256":"8b4044405da098f7a2fa33921ba378e8f3dd271118f47d6df95a94d6730f7a0b","abstract_canon_sha256":"b03a594bcd17a09e22c27f06c60b3d574e58b6c4c9add0c0be512414dfe32acc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:04.437552Z","signature_b64":"qDFmIYlg0Jb0R28p1K1bciwVxuGy+8AYJBVUiFcVcUcPY4c0zeaMZEylcPd2k+0wNDHlNFXdcBrVbeyi5CwgBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9ecafaf2d8bc881c588e2e604d5955c12992aa107ab4129773f26e77a06c0b1","last_reissued_at":"2026-05-18T00:44:04.436918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:04.436918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting De Bruijn sequences as perturbations of linear recursions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Don Coppersmith, Jeffrey M. VanderKam, Robert C. Rhoades","submitted_at":"2017-05-22T16:29:08Z","abstract_excerpt":"Every binary De~Bruijn sequence of order n satisfies a recursion 0=x_n+x_0+g(x_{n-1}, ..., x_1). Given a function f on (n-1) bits, let N(f; r) be the number of functions generating a De Bruijn sequence of order n which are obtained by changing r locations in the truth table of f. We prove a formula for the generating function \\sum_r N(\\ell; r) y^r when \\ell is a linear function.\n  The proof uses a weighted Matrix Tree Theorem and a description of the in-trees (or rooted trees) in the n-bit De Bruijn graph as perturbations of the Hamiltonian paths in the same graph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.07835","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1705.07835","created_at":"2026-05-18T00:44:04.437010+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.07835v1","created_at":"2026-05-18T00:44:04.437010+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.07835","created_at":"2026-05-18T00:44:04.437010+00:00"},{"alias_kind":"pith_short_12","alias_value":"3HWK7LZNRPEI","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"3HWK7LZNRPEIDRMI","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"3HWK7LZN","created_at":"2026-05-18T12:30:58.224056+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ","json":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ.json","graph_json":"https://pith.science/api/pith-number/3HWK7LZNRPEIDRMI4LTAJVMVLQ/graph.json","events_json":"https://pith.science/api/pith-number/3HWK7LZNRPEIDRMI4LTAJVMVLQ/events.json","paper":"https://pith.science/paper/3HWK7LZN"},"agent_actions":{"view_html":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ","download_json":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ.json","view_paper":"https://pith.science/paper/3HWK7LZN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.07835&json=true","fetch_graph":"https://pith.science/api/pith-number/3HWK7LZNRPEIDRMI4LTAJVMVLQ/graph.json","fetch_events":"https://pith.science/api/pith-number/3HWK7LZNRPEIDRMI4LTAJVMVLQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ/action/storage_attestation","attest_author":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ/action/author_attestation","sign_citation":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ/action/citation_signature","submit_replication":"https://pith.science/pith/3HWK7LZNRPEIDRMI4LTAJVMVLQ/action/replication_record"}},"created_at":"2026-05-18T00:44:04.437010+00:00","updated_at":"2026-05-18T00:44:04.437010+00:00"}