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It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free language to compute the generating function: \\begin{align*} \\sum_{n\\geq 0} c_n t^n &= \\frac{(1+q)\\left(1+5q-q^2-q^3-(1-q)\\sqrt{(1-q^2)(1-4q-q^2)}\\right)}{8q} \\end{al"},"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":"1407.4248","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-16T10:17:53Z","cross_cats_sorted":[],"title_canon_sha256":"b060d4d05eb782e866e72682942d5502f5a8f42c1510b488d7cb26dd60a4c45f","abstract_canon_sha256":"a84590f82ed48cc9c88fcd8567390ab1b896bfd4a3c1873737b08559d507315b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:57.568871Z","signature_b64":"rn1KIkGIoO4RiL23KlpbAO0es+LHOTIilsdcntPgXTA1pmScUAmki8EPE/s5aFnXOOXNpD+xcqzK58E0ZqoJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dec0040561614a156f8eb2cf9944bc6adc5e425a7600a21d8201d4908d18438","last_reissued_at":"2026-05-18T02:45:57.568336Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:57.568336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutations generated by a depth 2 and infinite stack in series are algebraic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Rechnitzer, Geoffrey Lee, Murray Elder","submitted_at":"2014-07-16T10:17:53Z","abstract_excerpt":"We prove that the class of permutations generated by passing an ordered sequence $12\\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free language to compute the generating function: \\begin{align*} \\sum_{n\\geq 0} c_n t^n &= \\frac{(1+q)\\left(1+5q-q^2-q^3-(1-q)\\sqrt{(1-q^2)(1-4q-q^2)}\\right)}{8q} \\end{al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4248","kind":"arxiv","version":2},"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":"1407.4248","created_at":"2026-05-18T02:45:57.568418+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.4248v2","created_at":"2026-05-18T02:45:57.568418+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.4248","created_at":"2026-05-18T02:45:57.568418+00:00"},{"alias_kind":"pith_short_12","alias_value":"NXWAAQCWCYKK","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"NXWAAQCWCYKKCVXY","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"NXWAAQCW","created_at":"2026-05-18T12:28:41.024544+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/NXWAAQCWCYKKCVXY5MWPTFCLY2","json":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2.json","graph_json":"https://pith.science/api/pith-number/NXWAAQCWCYKKCVXY5MWPTFCLY2/graph.json","events_json":"https://pith.science/api/pith-number/NXWAAQCWCYKKCVXY5MWPTFCLY2/events.json","paper":"https://pith.science/paper/NXWAAQCW"},"agent_actions":{"view_html":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2","download_json":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2.json","view_paper":"https://pith.science/paper/NXWAAQCW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.4248&json=true","fetch_graph":"https://pith.science/api/pith-number/NXWAAQCWCYKKCVXY5MWPTFCLY2/graph.json","fetch_events":"https://pith.science/api/pith-number/NXWAAQCWCYKKCVXY5MWPTFCLY2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2/action/storage_attestation","attest_author":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2/action/author_attestation","sign_citation":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2/action/citation_signature","submit_replication":"https://pith.science/pith/NXWAAQCWCYKKCVXY5MWPTFCLY2/action/replication_record"}},"created_at":"2026-05-18T02:45:57.568418+00:00","updated_at":"2026-05-18T02:45:57.568418+00:00"}