{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FYPLYQXUYBSBCLAPQYBAJMUWCH","short_pith_number":"pith:FYPLYQXU","canonical_record":{"source":{"id":"1712.05747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-15T16:50:32Z","cross_cats_sorted":[],"title_canon_sha256":"7dc18ec5201ac7f87dbe611455354609daf5ff4e892b36016562a612f9596e45","abstract_canon_sha256":"5c50941ee508c53545b848237ce27b5644b9b2ad4fd30e9b36055d9e0d20b52e"},"schema_version":"1.0"},"canonical_sha256":"2e1ebc42f4c064112c0f860204b29611dd4738548ec7f5040c02896cadbf9f28","source":{"kind":"arxiv","id":"1712.05747","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.05747","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1712.05747v1","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.05747","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"FYPLYQXUYBSB","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FYPLYQXUYBSBCLAP","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FYPLYQXU","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FYPLYQXUYBSBCLAPQYBAJMUWCH","target":"record","payload":{"canonical_record":{"source":{"id":"1712.05747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-15T16:50:32Z","cross_cats_sorted":[],"title_canon_sha256":"7dc18ec5201ac7f87dbe611455354609daf5ff4e892b36016562a612f9596e45","abstract_canon_sha256":"5c50941ee508c53545b848237ce27b5644b9b2ad4fd30e9b36055d9e0d20b52e"},"schema_version":"1.0"},"canonical_sha256":"2e1ebc42f4c064112c0f860204b29611dd4738548ec7f5040c02896cadbf9f28","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:55.779945Z","signature_b64":"uYZ5v69mRbafYMHGbQQOGQIekIjau9PQGl1S50Sz8ju3sUkjE2psUj8PWQ7W1C/UUf4BHsAys/bp85y3G1w5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e1ebc42f4c064112c0f860204b29611dd4738548ec7f5040c02896cadbf9f28","last_reissued_at":"2026-05-18T00:27:55.779333Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:55.779333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.05747","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pXMes/S1BmW2xddFD/lBo2+cNJD0PEV11USpXuohNQUIZCIBvt+LRx/q9l0cJ6XQyKUmy9KbfwyhpIDv9qssCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T21:46:31.050853Z"},"content_sha256":"e583c5d4cb32a1074926a45d4406301987c4ca79a0b6581c052ce51749c4990a","schema_version":"1.0","event_id":"sha256:e583c5d4cb32a1074926a45d4406301987c4ca79a0b6581c052ce51749c4990a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FYPLYQXUYBSBCLAPQYBAJMUWCH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Hilbert series of the Grassmannian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Lukas Braun","submitted_at":"2017-12-15T16:50:32Z","abstract_excerpt":"We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods and show that its h-polynomial coincides with the k-Narayana polynomial. We give a simplified formula for the h-polynomial of Schubert varieties. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k-Narayana numbers, i.e. the h-polynomial of the Grassmannian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.05747","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ve9UjpOr/brvBe5xCRkJvmaqSsIRk7aozkpMtzWfXgDn+WZrrLSGyiiALJZAzu3aYPpBMQ/Ea3q62HnAGdsaDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T21:46:31.051232Z"},"content_sha256":"b80c44bb399c32ed48eee5e6e148ad8bfd6dfcf393924e276097c2c18c91ea83","schema_version":"1.0","event_id":"sha256:b80c44bb399c32ed48eee5e6e148ad8bfd6dfcf393924e276097c2c18c91ea83"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/bundle.json","state_url":"https://pith.science/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/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":"2026-05-29T21:46:31Z","links":{"resolver":"https://pith.science/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH","bundle":"https://pith.science/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/bundle.json","state":"https://pith.science/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FYPLYQXUYBSBCLAPQYBAJMUWCH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FYPLYQXUYBSBCLAPQYBAJMUWCH","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":"5c50941ee508c53545b848237ce27b5644b9b2ad4fd30e9b36055d9e0d20b52e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-15T16:50:32Z","title_canon_sha256":"7dc18ec5201ac7f87dbe611455354609daf5ff4e892b36016562a612f9596e45"},"schema_version":"1.0","source":{"id":"1712.05747","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.05747","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1712.05747v1","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.05747","created_at":"2026-05-18T00:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"FYPLYQXUYBSB","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FYPLYQXUYBSBCLAP","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FYPLYQXU","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:b80c44bb399c32ed48eee5e6e148ad8bfd6dfcf393924e276097c2c18c91ea83","target":"graph","created_at":"2026-05-18T00:27:55Z","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 compute the Hilbert series of the complex Grassmannian using invariant theoretic methods and show that its h-polynomial coincides with the k-Narayana polynomial. We give a simplified formula for the h-polynomial of Schubert varieties. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k-Narayana numbers, i.e. the h-polynomial of the Grassmannian.","authors_text":"Lukas Braun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-15T16:50:32Z","title":"On the Hilbert series of the Grassmannian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.05747","kind":"arxiv","version":1},"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:e583c5d4cb32a1074926a45d4406301987c4ca79a0b6581c052ce51749c4990a","target":"record","created_at":"2026-05-18T00:27:55Z","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":"5c50941ee508c53545b848237ce27b5644b9b2ad4fd30e9b36055d9e0d20b52e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-12-15T16:50:32Z","title_canon_sha256":"7dc18ec5201ac7f87dbe611455354609daf5ff4e892b36016562a612f9596e45"},"schema_version":"1.0","source":{"id":"1712.05747","kind":"arxiv","version":1}},"canonical_sha256":"2e1ebc42f4c064112c0f860204b29611dd4738548ec7f5040c02896cadbf9f28","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2e1ebc42f4c064112c0f860204b29611dd4738548ec7f5040c02896cadbf9f28","first_computed_at":"2026-05-18T00:27:55.779333Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:55.779333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uYZ5v69mRbafYMHGbQQOGQIekIjau9PQGl1S50Sz8ju3sUkjE2psUj8PWQ7W1C/UUf4BHsAys/bp85y3G1w5Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:55.779945Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.05747","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e583c5d4cb32a1074926a45d4406301987c4ca79a0b6581c052ce51749c4990a","sha256:b80c44bb399c32ed48eee5e6e148ad8bfd6dfcf393924e276097c2c18c91ea83"],"state_sha256":"73827db002d680aa164cb322340a3797eebac51d632175b4ac306644ca224cea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2e0wRJnmnVdxHZ3KyTRhWIlUnYSsr2zdvgc3ojJbl5wzWxzqamzUOvDeQsmUwSDVbh34DLxmm7f8o0v+cNT9BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T21:46:31.053361Z","bundle_sha256":"4a929eb0a37c925bb075f511fc45765aaf11f73ef8ca9e8d25ebed6c0dcea207"}}