{"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"}