{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:MCCGFY555FFGR3NONNOHLFEY3N","short_pith_number":"pith:MCCGFY55","canonical_record":{"source":{"id":"1009.2935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-09-15T14:28:45Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"978a9c5fa63e3ab67bc1afcebc2119ff8cf8099655b83e6fed4a08fbf3c16988","abstract_canon_sha256":"ab8a5c2f67f9f512413fbad04d4749d5ad20dc5f3a470e7fff98a34bdb41a8ba"},"schema_version":"1.0"},"canonical_sha256":"608462e3bde94a68edae6b5c759498db732592a490241a5ff718b8037d77ee3a","source":{"kind":"arxiv","id":"1009.2935","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2935","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2935v2","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2935","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"pith_short_12","alias_value":"MCCGFY555FFG","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MCCGFY555FFGR3NO","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MCCGFY55","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:MCCGFY555FFGR3NONNOHLFEY3N","target":"record","payload":{"canonical_record":{"source":{"id":"1009.2935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-09-15T14:28:45Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"978a9c5fa63e3ab67bc1afcebc2119ff8cf8099655b83e6fed4a08fbf3c16988","abstract_canon_sha256":"ab8a5c2f67f9f512413fbad04d4749d5ad20dc5f3a470e7fff98a34bdb41a8ba"},"schema_version":"1.0"},"canonical_sha256":"608462e3bde94a68edae6b5c759498db732592a490241a5ff718b8037d77ee3a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:02.276569Z","signature_b64":"qVaqyi8ym7luF9u+8YGoNoQWJP+5CDz7KCeZJU7C9Xnmi/6ZBBcFt03yx4KFlSFwT8cTgR+DCDKFTIfc2BMRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"608462e3bde94a68edae6b5c759498db732592a490241a5ff718b8037d77ee3a","last_reissued_at":"2026-05-18T04:12:02.276185Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:02.276185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.2935","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"69XG+3tqXfqyXxJGZmTuRZy1zJI1+niPIvdIKfh9wz3c73wd4ZmCa71+ti0r4hzePoomFgO6jNGlxW31IF8LCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:56:49.754716Z"},"content_sha256":"d0bbe80a9924bcf50338b777e2141c871470dbbaf00b26500a27a6dcff4570dd","schema_version":"1.0","event_id":"sha256:d0bbe80a9924bcf50338b777e2141c871470dbbaf00b26500a27a6dcff4570dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:MCCGFY555FFGR3NONNOHLFEY3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discretized configurations and partial partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.GT","authors_text":"Aaron Abrams, David Gay, Valerie Hower","submitted_at":"2010-09-15T14:28:45Z","abstract_excerpt":"We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\\{1,...,n+1\\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2935","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1mEGl2Z5NvTDoM4GGL27vXyACFnR6HTrKHSfTdNseRR8wIo0Z9MSl6QXD5ZKDXj1s51y6WkHlSz4K1yWUHUdDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:56:49.755057Z"},"content_sha256":"98939a9a0cd0d5fdf0fb8b23e1740178c9297ce5edea342e441cddb59229a1d7","schema_version":"1.0","event_id":"sha256:98939a9a0cd0d5fdf0fb8b23e1740178c9297ce5edea342e441cddb59229a1d7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MCCGFY555FFGR3NONNOHLFEY3N/bundle.json","state_url":"https://pith.science/pith/MCCGFY555FFGR3NONNOHLFEY3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MCCGFY555FFGR3NONNOHLFEY3N/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":"2026-06-02T07:56:49Z","links":{"resolver":"https://pith.science/pith/MCCGFY555FFGR3NONNOHLFEY3N","bundle":"https://pith.science/pith/MCCGFY555FFGR3NONNOHLFEY3N/bundle.json","state":"https://pith.science/pith/MCCGFY555FFGR3NONNOHLFEY3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MCCGFY555FFGR3NONNOHLFEY3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MCCGFY555FFGR3NONNOHLFEY3N","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":"ab8a5c2f67f9f512413fbad04d4749d5ad20dc5f3a470e7fff98a34bdb41a8ba","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-09-15T14:28:45Z","title_canon_sha256":"978a9c5fa63e3ab67bc1afcebc2119ff8cf8099655b83e6fed4a08fbf3c16988"},"schema_version":"1.0","source":{"id":"1009.2935","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.2935","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"arxiv_version","alias_value":"1009.2935v2","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.2935","created_at":"2026-05-18T04:12:02Z"},{"alias_kind":"pith_short_12","alias_value":"MCCGFY555FFG","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MCCGFY555FFGR3NO","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MCCGFY55","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:98939a9a0cd0d5fdf0fb8b23e1740178c9297ce5edea342e441cddb59229a1d7","target":"graph","created_at":"2026-05-18T04:12:02Z","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":"We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\\{1,...,n+1\\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.","authors_text":"Aaron Abrams, David Gay, Valerie Hower","cross_cats":["math.AT","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-09-15T14:28:45Z","title":"Discretized configurations and partial partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2935","kind":"arxiv","version":2},"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:d0bbe80a9924bcf50338b777e2141c871470dbbaf00b26500a27a6dcff4570dd","target":"record","created_at":"2026-05-18T04:12:02Z","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":"ab8a5c2f67f9f512413fbad04d4749d5ad20dc5f3a470e7fff98a34bdb41a8ba","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-09-15T14:28:45Z","title_canon_sha256":"978a9c5fa63e3ab67bc1afcebc2119ff8cf8099655b83e6fed4a08fbf3c16988"},"schema_version":"1.0","source":{"id":"1009.2935","kind":"arxiv","version":2}},"canonical_sha256":"608462e3bde94a68edae6b5c759498db732592a490241a5ff718b8037d77ee3a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"608462e3bde94a68edae6b5c759498db732592a490241a5ff718b8037d77ee3a","first_computed_at":"2026-05-18T04:12:02.276185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:02.276185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qVaqyi8ym7luF9u+8YGoNoQWJP+5CDz7KCeZJU7C9Xnmi/6ZBBcFt03yx4KFlSFwT8cTgR+DCDKFTIfc2BMRDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:02.276569Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.2935","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0bbe80a9924bcf50338b777e2141c871470dbbaf00b26500a27a6dcff4570dd","sha256:98939a9a0cd0d5fdf0fb8b23e1740178c9297ce5edea342e441cddb59229a1d7"],"state_sha256":"19c2c4de96fb0f3145adde589452e208b5c5fbcaf663e64471d47c07a1eac2a1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dYYG5h3j8lI2NPHjVpyhwDev2lNyaZTXWuFjmjSnz4S9JNGGxFrwOdetE4P5qZSUY8nQLQv4sMV58OcpTtotCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T07:56:49.757011Z","bundle_sha256":"dc8e5a79f07ec7b418bd2b76f32328a8111b0fc572787e7f025f113d3d055cf5"}}