{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3L3KUTAY4VTMPBW5IH5HPOMTK4","short_pith_number":"pith:3L3KUTAY","canonical_record":{"source":{"id":"1609.01782","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T22:50:54Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8c4e83d4a86b46139533d0b203630b357888cb1c592066de2aca2e83d13574aa","abstract_canon_sha256":"3d26eb22a2905d2acc5f9eca6f1fde0951003092f6724154bfa78dc6fb9080d7"},"schema_version":"1.0"},"canonical_sha256":"daf6aa4c18e566c786dd41fa77b99357313007639e488b05d434748fc58751b9","source":{"kind":"arxiv","id":"1609.01782","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01782","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01782v3","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01782","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"3L3KUTAY4VTM","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3L3KUTAY4VTMPBW5","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3L3KUTAY","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3L3KUTAY4VTMPBW5IH5HPOMTK4","target":"record","payload":{"canonical_record":{"source":{"id":"1609.01782","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T22:50:54Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8c4e83d4a86b46139533d0b203630b357888cb1c592066de2aca2e83d13574aa","abstract_canon_sha256":"3d26eb22a2905d2acc5f9eca6f1fde0951003092f6724154bfa78dc6fb9080d7"},"schema_version":"1.0"},"canonical_sha256":"daf6aa4c18e566c786dd41fa77b99357313007639e488b05d434748fc58751b9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:41.686905Z","signature_b64":"qiR96HleW+TO1dZxja3lJFiBj+iUBguzQ05nREoS+ipdqwLkpv+ImRgFT7VWjO+gO78046VaRgpGkpSEkjn6Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daf6aa4c18e566c786dd41fa77b99357313007639e488b05d434748fc58751b9","last_reissued_at":"2026-05-18T00:10:41.686238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:41.686238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.01782","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F4IAjSE49epNdEJ4l1OIvT3gGannQgj4vphKEG7DnhWFohovRfg/5vo4cczuVu1nEqr9/cF5iq1h9HGE0j8vDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:58:21.756059Z"},"content_sha256":"a627a8de90383f26520df99412fe8eec6d372b5d110e8e3d6d16fb8c2e71e966","schema_version":"1.0","event_id":"sha256:a627a8de90383f26520df99412fe8eec6d372b5d110e8e3d6d16fb8c2e71e966"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3L3KUTAY4VTMPBW5IH5HPOMTK4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pattern-Avoiding Polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Bruce Sagan, Robert Davis","submitted_at":"2016-09-06T22:50:54Z","abstract_excerpt":"Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(\\Pi)$ and $B_n(\\Pi)$, whose vertices correspond to the permutations in $\\mathfrak{S}_n$ avoiding a set of patterns $\\Pi$. For various choices of $\\Pi$, we explore the Ehrhart polynomials and $h^*$-vectors of these polytopes as well as other aspects of their combinatorial structure.\n  For $P_n(\\Pi)$, we consider all subsets $\\Pi \\subseteq \\mathfrak{S}_3$ and are able to provide results in most cases. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01782","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tu24M7zzyyl6XcWbcaiUN9qCVGAdwEopikZE2Aomsbhzayg+VO/rCpWz17kj5xsSK4h5PBrk1NwV+OUowAXRBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:58:21.756428Z"},"content_sha256":"ac5749d9248c276c66dda915182d51a7a36b24a359160a92349722c4285d04c6","schema_version":"1.0","event_id":"sha256:ac5749d9248c276c66dda915182d51a7a36b24a359160a92349722c4285d04c6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/bundle.json","state_url":"https://pith.science/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T22:58:21Z","links":{"resolver":"https://pith.science/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4","bundle":"https://pith.science/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/bundle.json","state":"https://pith.science/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3L3KUTAY4VTMPBW5IH5HPOMTK4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3L3KUTAY4VTMPBW5IH5HPOMTK4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3d26eb22a2905d2acc5f9eca6f1fde0951003092f6724154bfa78dc6fb9080d7","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T22:50:54Z","title_canon_sha256":"8c4e83d4a86b46139533d0b203630b357888cb1c592066de2aca2e83d13574aa"},"schema_version":"1.0","source":{"id":"1609.01782","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01782","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01782v3","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01782","created_at":"2026-05-18T00:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"3L3KUTAY4VTM","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3L3KUTAY4VTMPBW5","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3L3KUTAY","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:ac5749d9248c276c66dda915182d51a7a36b24a359160a92349722c4285d04c6","target":"graph","created_at":"2026-05-18T00:10:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(\\Pi)$ and $B_n(\\Pi)$, whose vertices correspond to the permutations in $\\mathfrak{S}_n$ avoiding a set of patterns $\\Pi$. For various choices of $\\Pi$, we explore the Ehrhart polynomials and $h^*$-vectors of these polytopes as well as other aspects of their combinatorial structure.\n  For $P_n(\\Pi)$, we consider all subsets $\\Pi \\subseteq \\mathfrak{S}_3$ and are able to provide results in most cases. T","authors_text":"Bruce Sagan, Robert Davis","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T22:50:54Z","title":"Pattern-Avoiding Polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01782","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a627a8de90383f26520df99412fe8eec6d372b5d110e8e3d6d16fb8c2e71e966","target":"record","created_at":"2026-05-18T00:10:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3d26eb22a2905d2acc5f9eca6f1fde0951003092f6724154bfa78dc6fb9080d7","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-06T22:50:54Z","title_canon_sha256":"8c4e83d4a86b46139533d0b203630b357888cb1c592066de2aca2e83d13574aa"},"schema_version":"1.0","source":{"id":"1609.01782","kind":"arxiv","version":3}},"canonical_sha256":"daf6aa4c18e566c786dd41fa77b99357313007639e488b05d434748fc58751b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"daf6aa4c18e566c786dd41fa77b99357313007639e488b05d434748fc58751b9","first_computed_at":"2026-05-18T00:10:41.686238Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:41.686238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qiR96HleW+TO1dZxja3lJFiBj+iUBguzQ05nREoS+ipdqwLkpv+ImRgFT7VWjO+gO78046VaRgpGkpSEkjn6Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:41.686905Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.01782","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a627a8de90383f26520df99412fe8eec6d372b5d110e8e3d6d16fb8c2e71e966","sha256:ac5749d9248c276c66dda915182d51a7a36b24a359160a92349722c4285d04c6"],"state_sha256":"5ca1e6051938783db961a1fcc85e57021627dc9274be95d2b6413f3f79b415b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8UfhY8k40fImP6HU6djONM/Zs93ruxdt+RRhk9PyWVPvTG8bqtVtsiIivatF1XWuLO65rcuqUaeciaSD4XTLCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T22:58:21.758457Z","bundle_sha256":"5bbffd6b10b39d6cdd4ac51e8b885c8449b64c5c4f85bd3fba9016ad7cf6282b"}}