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We then show that this implies that for most patterns $q$, the generating function $\\sum_{n\\geq 0} \\textup{Av}_n(q)z^n$ of the sequence $\\textup{Av}_n(q)$ of the numbers of $q$-avoiding permutations is not rational. Placing our results in a broader context, we show that for rational power series $F(z)$ and $G(z)$ with nonnegative real coefficients, the relation $F(z"},"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":"1901.08506","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-24T17:03:07Z","cross_cats_sorted":[],"title_canon_sha256":"e151ef16f9bb553423e0f0dcbdcf01a4829615489c8f69c5020d095ac39bd910","abstract_canon_sha256":"396ec985c01ba42e780b36d1faf9b967ee08b3ebc5b673999072865f691cbfd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:29.517828Z","signature_b64":"a3DTFg6p3m+D6XnwMyxCBQf3C0f47p/rFNQZiHOTdCMomhiYjyIVGr4LfOE2nYwTVLtFYBS2ebhrjXCQj8n+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5b7b930f49bc406d87f5313fa667752b8d9019021a5ad5a9884d3f630bc6578","last_reissued_at":"2026-05-17T23:44:29.517146Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:29.517146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Most principal permutation classes have nonrational generating functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mikl\\'os B\\'ona","submitted_at":"2019-01-24T17:03:07Z","abstract_excerpt":"We prove that for any fixed $n$, and for most permutation patterns $q$, the number $\\textup{Av}_{n,\\ell}(q)$ of $q$-avoiding permutations of length $n$ that consist of $\\ell$ skew blocks is a monotone decreasing function of $\\ell$. We then show that this implies that for most patterns $q$, the generating function $\\sum_{n\\geq 0} \\textup{Av}_n(q)z^n$ of the sequence $\\textup{Av}_n(q)$ of the numbers of $q$-avoiding permutations is not rational. Placing our results in a broader context, we show that for rational power series $F(z)$ and $G(z)$ with nonnegative real coefficients, the relation $F(z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.08506","kind":"arxiv","version":4},"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":"1901.08506","created_at":"2026-05-17T23:44:29.517261+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.08506v4","created_at":"2026-05-17T23:44:29.517261+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.08506","created_at":"2026-05-17T23:44:29.517261+00:00"},{"alias_kind":"pith_short_12","alias_value":"UW33SMHUTPCA","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"UW33SMHUTPCANWD7","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"UW33SMHU","created_at":"2026-05-18T12:33:30.264802+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/UW33SMHUTPCANWD7KMJ7UZTXKK","json":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK.json","graph_json":"https://pith.science/api/pith-number/UW33SMHUTPCANWD7KMJ7UZTXKK/graph.json","events_json":"https://pith.science/api/pith-number/UW33SMHUTPCANWD7KMJ7UZTXKK/events.json","paper":"https://pith.science/paper/UW33SMHU"},"agent_actions":{"view_html":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK","download_json":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK.json","view_paper":"https://pith.science/paper/UW33SMHU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.08506&json=true","fetch_graph":"https://pith.science/api/pith-number/UW33SMHUTPCANWD7KMJ7UZTXKK/graph.json","fetch_events":"https://pith.science/api/pith-number/UW33SMHUTPCANWD7KMJ7UZTXKK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK/action/storage_attestation","attest_author":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK/action/author_attestation","sign_citation":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK/action/citation_signature","submit_replication":"https://pith.science/pith/UW33SMHUTPCANWD7KMJ7UZTXKK/action/replication_record"}},"created_at":"2026-05-17T23:44:29.517261+00:00","updated_at":"2026-05-17T23:44:29.517261+00:00"}