{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:54KU7E4JAOXJXCQRM275KILDDQ","short_pith_number":"pith:54KU7E4J","canonical_record":{"source":{"id":"1306.6910","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-28T17:54:19Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"21736f38cb3ac234a8a0c37de67f79328028b97ce6df220ad643d98eb4ab7388","abstract_canon_sha256":"0f3969c353db9b07e475739b487065b9fc37b76d6add8b562a50b1cebdc29020"},"schema_version":"1.0"},"canonical_sha256":"ef154f938903ae9b8a1166bfd521631c23906c55d00564f167d5d6eddbc653d2","source":{"kind":"arxiv","id":"1306.6910","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6910","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6910v2","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6910","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"pith_short_12","alias_value":"54KU7E4JAOXJ","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"54KU7E4JAOXJXCQR","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"54KU7E4J","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:54KU7E4JAOXJXCQRM275KILDDQ","target":"record","payload":{"canonical_record":{"source":{"id":"1306.6910","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-28T17:54:19Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"21736f38cb3ac234a8a0c37de67f79328028b97ce6df220ad643d98eb4ab7388","abstract_canon_sha256":"0f3969c353db9b07e475739b487065b9fc37b76d6add8b562a50b1cebdc29020"},"schema_version":"1.0"},"canonical_sha256":"ef154f938903ae9b8a1166bfd521631c23906c55d00564f167d5d6eddbc653d2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:58.263894Z","signature_b64":"bypA6yOBWKTS7UqapfDAMzxUZ2ReWuxxgyhMSglBDoOSL1orwtFpurR2PKGHm473hd77MYHM9ey4cwHlTUivDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef154f938903ae9b8a1166bfd521631c23906c55d00564f167d5d6eddbc653d2","last_reissued_at":"2026-05-18T03:01:58.263271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:58.263271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.6910","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":"2026-05-18T03:01:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5As4TwM4PeKIAIok+YxiV5vyGDi2Fw2Vq7Ad8R8PVas7kzm8DlNi5thsim+Wto7VRZW7IUaoxZuj8+lQVX54Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:37:43.028242Z"},"content_sha256":"4bdfb813777ed83ea29a56767af580b430dadc51a60657de48b3b40bd5a797f9","schema_version":"1.0","event_id":"sha256:4bdfb813777ed83ea29a56767af580b430dadc51a60657de48b3b40bd5a797f9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:54KU7E4JAOXJXCQRM275KILDDQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Segre embeddings, Hilbert series and Newcomb's problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Marcel Morales (IF)","submitted_at":"2013-06-28T17:54:19Z","abstract_excerpt":"Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\\rm in}_\\prec I$, in some special situations the monomial ideal ${\\rm in}_\\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original questio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6910","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":"2026-05-18T03:01:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Svtmemphcq0ol3hyQy3cNCfItvcNDR3xo+w35SDitqWd9ZEjYaa18HNBYy5HPveRJ+F+JeXeGr/tQAdIkdRKDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:37:43.028592Z"},"content_sha256":"9ae3656f826f7fc11d995674b87ce27387ff3f4ffa95d2c8aa69dda83191736d","schema_version":"1.0","event_id":"sha256:9ae3656f826f7fc11d995674b87ce27387ff3f4ffa95d2c8aa69dda83191736d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/54KU7E4JAOXJXCQRM275KILDDQ/bundle.json","state_url":"https://pith.science/pith/54KU7E4JAOXJXCQRM275KILDDQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/54KU7E4JAOXJXCQRM275KILDDQ/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":"2026-06-27T17:37:43Z","links":{"resolver":"https://pith.science/pith/54KU7E4JAOXJXCQRM275KILDDQ","bundle":"https://pith.science/pith/54KU7E4JAOXJXCQRM275KILDDQ/bundle.json","state":"https://pith.science/pith/54KU7E4JAOXJXCQRM275KILDDQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/54KU7E4JAOXJXCQRM275KILDDQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:54KU7E4JAOXJXCQRM275KILDDQ","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":"0f3969c353db9b07e475739b487065b9fc37b76d6add8b562a50b1cebdc29020","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-28T17:54:19Z","title_canon_sha256":"21736f38cb3ac234a8a0c37de67f79328028b97ce6df220ad643d98eb4ab7388"},"schema_version":"1.0","source":{"id":"1306.6910","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6910","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6910v2","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6910","created_at":"2026-05-18T03:01:58Z"},{"alias_kind":"pith_short_12","alias_value":"54KU7E4JAOXJ","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"54KU7E4JAOXJXCQR","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"54KU7E4J","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:9ae3656f826f7fc11d995674b87ce27387ff3f4ffa95d2c8aa69dda83191736d","target":"graph","created_at":"2026-05-18T03:01:58Z","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":"Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\\rm in}_\\prec I$, in some special situations the monomial ideal ${\\rm in}_\\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original questio","authors_text":"Marcel Morales (IF)","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-28T17:54:19Z","title":"Segre embeddings, Hilbert series and Newcomb's problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6910","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":"sha256:4bdfb813777ed83ea29a56767af580b430dadc51a60657de48b3b40bd5a797f9","target":"record","created_at":"2026-05-18T03:01:58Z","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":"0f3969c353db9b07e475739b487065b9fc37b76d6add8b562a50b1cebdc29020","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-06-28T17:54:19Z","title_canon_sha256":"21736f38cb3ac234a8a0c37de67f79328028b97ce6df220ad643d98eb4ab7388"},"schema_version":"1.0","source":{"id":"1306.6910","kind":"arxiv","version":2}},"canonical_sha256":"ef154f938903ae9b8a1166bfd521631c23906c55d00564f167d5d6eddbc653d2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef154f938903ae9b8a1166bfd521631c23906c55d00564f167d5d6eddbc653d2","first_computed_at":"2026-05-18T03:01:58.263271Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:01:58.263271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bypA6yOBWKTS7UqapfDAMzxUZ2ReWuxxgyhMSglBDoOSL1orwtFpurR2PKGHm473hd77MYHM9ey4cwHlTUivDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:01:58.263894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.6910","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4bdfb813777ed83ea29a56767af580b430dadc51a60657de48b3b40bd5a797f9","sha256:9ae3656f826f7fc11d995674b87ce27387ff3f4ffa95d2c8aa69dda83191736d"],"state_sha256":"6bba7c4162564ec0f687204db98ecd036ec27cfcadc3c201e8f126c909d7870e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q5lpWVf7pzZ6gzs0BlmOxciP+FkYBLZLvyiiu3C4/y+DWA5NWzCa/xgZ595yMdP6GM9S9oWtrn2K+fBbo8g2Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T17:37:43.030511Z","bundle_sha256":"f38efe6a9b39d5d07a460c614c077062c9d8bea9801f9e5878e646cd5b05e584"}}