{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:EKO2QVKWWRWVYH7JUAMD6BHHUO","short_pith_number":"pith:EKO2QVKW","schema_version":"1.0","canonical_sha256":"229da85556b46d5c1fe9a0183f04e7a38b8f98f930b9151a870ea68ec5503622","source":{"kind":"arxiv","id":"1702.08125","version":1},"attestation_state":"computed","paper":{"title":"Generating functions for permutations which avoid consecutive patterns with multiple descents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeffrey B. Remmel, Quang T. Bach","submitted_at":"2017-02-27T02:14:37Z","abstract_excerpt":"Let $S_n$ denote the group all permutations of $n$. For every permutation $\\sigma$, we let $\\mathrm{des}(\\sigma)$ denote the number of descents in $\\sigma$ and $\\mathrm{LRMin}(\\sigma)$ denote the number of left-to-right minima of $\\sigma$. Given a sequence $\\tau = \\tau_1 \\cdots \\tau_n$ of distinct positive integers, we define the reduction of $\\tau$, $\\mathrm{red}(\\tau)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $\\tau$ by $i$.\n  If $\\Gamma$ is a set of permutations, we say that a permutation $\\sigma = \\sigma_1 \\ldots \\sigma_n \\in S_n$ has a $\\Gamm"},"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":"1702.08125","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-27T02:14:37Z","cross_cats_sorted":[],"title_canon_sha256":"47dc076cd1201c2ef64afda8c2d3717bcf7c893fbd7111abb90942ef5a5b48b6","abstract_canon_sha256":"01a07c21c71a553ff50d457490be0abf1ea00859d71d9dcfb173629d8f97581e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:55.922705Z","signature_b64":"DPSEcUl5pTSGyW6LI8PskjBeZqVPUFisUJQYWi2vZUm7peZQhJLwCrhriM1rNtCFZmykz2fHtR1JmB7bgVoHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"229da85556b46d5c1fe9a0183f04e7a38b8f98f930b9151a870ea68ec5503622","last_reissued_at":"2026-05-18T00:49:55.921980Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:55.921980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generating functions for permutations which avoid consecutive patterns with multiple descents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeffrey B. Remmel, Quang T. Bach","submitted_at":"2017-02-27T02:14:37Z","abstract_excerpt":"Let $S_n$ denote the group all permutations of $n$. For every permutation $\\sigma$, we let $\\mathrm{des}(\\sigma)$ denote the number of descents in $\\sigma$ and $\\mathrm{LRMin}(\\sigma)$ denote the number of left-to-right minima of $\\sigma$. Given a sequence $\\tau = \\tau_1 \\cdots \\tau_n$ of distinct positive integers, we define the reduction of $\\tau$, $\\mathrm{red}(\\tau)$, to be the permutation of $S_n$ that results by replacing the $i$-th smallest element of $\\tau$ by $i$.\n  If $\\Gamma$ is a set of permutations, we say that a permutation $\\sigma = \\sigma_1 \\ldots \\sigma_n \\in S_n$ has a $\\Gamm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.08125","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":"1702.08125","created_at":"2026-05-18T00:49:55.922104+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.08125v1","created_at":"2026-05-18T00:49:55.922104+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.08125","created_at":"2026-05-18T00:49:55.922104+00:00"},{"alias_kind":"pith_short_12","alias_value":"EKO2QVKWWRWV","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"EKO2QVKWWRWVYH7J","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"EKO2QVKW","created_at":"2026-05-18T12:31:12.930513+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/EKO2QVKWWRWVYH7JUAMD6BHHUO","json":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO.json","graph_json":"https://pith.science/api/pith-number/EKO2QVKWWRWVYH7JUAMD6BHHUO/graph.json","events_json":"https://pith.science/api/pith-number/EKO2QVKWWRWVYH7JUAMD6BHHUO/events.json","paper":"https://pith.science/paper/EKO2QVKW"},"agent_actions":{"view_html":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO","download_json":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO.json","view_paper":"https://pith.science/paper/EKO2QVKW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.08125&json=true","fetch_graph":"https://pith.science/api/pith-number/EKO2QVKWWRWVYH7JUAMD6BHHUO/graph.json","fetch_events":"https://pith.science/api/pith-number/EKO2QVKWWRWVYH7JUAMD6BHHUO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO/action/storage_attestation","attest_author":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO/action/author_attestation","sign_citation":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO/action/citation_signature","submit_replication":"https://pith.science/pith/EKO2QVKWWRWVYH7JUAMD6BHHUO/action/replication_record"}},"created_at":"2026-05-18T00:49:55.922104+00:00","updated_at":"2026-05-18T00:49:55.922104+00:00"}