{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2HAUPVFCN6GJZZQSMWTIFKTGZE","short_pith_number":"pith:2HAUPVFC","canonical_record":{"source":{"id":"1311.4480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-18T18:26:38Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8b6aad0c48dd8c70beb148c809f44fb421acaec5cb41dfab962b8c3017214916","abstract_canon_sha256":"c09c119bf5c2ee5f99495b0c6edb2ca3a71d2d59c25c1bb9c392b4e16828c571"},"schema_version":"1.0"},"canonical_sha256":"d1c147d4a26f8c9ce61265a682aa66c93ae08d1cedf9415b3b516e6b6798fcb2","source":{"kind":"arxiv","id":"1311.4480","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4480","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4480v2","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4480","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"pith_short_12","alias_value":"2HAUPVFCN6GJ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"2HAUPVFCN6GJZZQS","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"2HAUPVFC","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2HAUPVFCN6GJZZQSMWTIFKTGZE","target":"record","payload":{"canonical_record":{"source":{"id":"1311.4480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-18T18:26:38Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8b6aad0c48dd8c70beb148c809f44fb421acaec5cb41dfab962b8c3017214916","abstract_canon_sha256":"c09c119bf5c2ee5f99495b0c6edb2ca3a71d2d59c25c1bb9c392b4e16828c571"},"schema_version":"1.0"},"canonical_sha256":"d1c147d4a26f8c9ce61265a682aa66c93ae08d1cedf9415b3b516e6b6798fcb2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:31.731983Z","signature_b64":"MgVUKJ2KBX4/3+6/QG7l8G4daUxxOhVJbZz/CZKCiUTxdtbGDsrwcm1KzT1zc80IBg/x8MyY1aR1waUvZ3ARBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1c147d4a26f8c9ce61265a682aa66c93ae08d1cedf9415b3b516e6b6798fcb2","last_reissued_at":"2026-05-18T02:18:31.731493Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:31.731493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.4480","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":"2026-05-18T02:18:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U5PDiMR0gKXqo7dBPcxS4PuVjROw0V+oMylaqkDVMc/oMoai/lnH7iFccI+pCfU8cOx/OnEaNiwajs6Yd9aBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T03:48:31.877577Z"},"content_sha256":"99510f3c91870e4c923e19e51d67418b974d9e3d2f3a0eddc3bf0cb92180b68e","schema_version":"1.0","event_id":"sha256:99510f3c91870e4c923e19e51d67418b974d9e3d2f3a0eddc3bf0cb92180b68e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2HAUPVFCN6GJZZQSMWTIFKTGZE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Zeilberger's KOH theorem and the strict unimodality of q-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Fabrizio Zanello","submitted_at":"2013-11-18T18:26:38Z","abstract_excerpt":"A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\\binom{a+b}{b}_q$ (see \\cite{PP} and also \\cite{PP2} for a slightly revised version of their theorem). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\\binom{a+b}{b}_q$ can get arbitrarily large"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4480","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":"2026-05-18T02:18:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+JPnxApaI8gbVzqqOgi09ljFV0a/x7/3C4XHvcYM6tExmly/C65ZcMHLpnraOwNB4YK52qm8gF5p+D4snb0LAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T03:48:31.877950Z"},"content_sha256":"65d466abd84207c605300661b29b55b73cab79bf76525632d46dfa213a09026a","schema_version":"1.0","event_id":"sha256:65d466abd84207c605300661b29b55b73cab79bf76525632d46dfa213a09026a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/bundle.json","state_url":"https://pith.science/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/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":"2026-06-09T03:48:31Z","links":{"resolver":"https://pith.science/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE","bundle":"https://pith.science/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/bundle.json","state":"https://pith.science/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2HAUPVFCN6GJZZQSMWTIFKTGZE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2HAUPVFCN6GJZZQSMWTIFKTGZE","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":"c09c119bf5c2ee5f99495b0c6edb2ca3a71d2d59c25c1bb9c392b4e16828c571","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-18T18:26:38Z","title_canon_sha256":"8b6aad0c48dd8c70beb148c809f44fb421acaec5cb41dfab962b8c3017214916"},"schema_version":"1.0","source":{"id":"1311.4480","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4480","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4480v2","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4480","created_at":"2026-05-18T02:18:31Z"},{"alias_kind":"pith_short_12","alias_value":"2HAUPVFCN6GJ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"2HAUPVFCN6GJZZQS","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"2HAUPVFC","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:65d466abd84207c605300661b29b55b73cab79bf76525632d46dfa213a09026a","target":"graph","created_at":"2026-05-18T02:18:31Z","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":"A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\\binom{a+b}{b}_q$ (see \\cite{PP} and also \\cite{PP2} for a slightly revised version of their theorem). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\\binom{a+b}{b}_q$ can get arbitrarily large","authors_text":"Fabrizio Zanello","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-18T18:26:38Z","title":"Zeilberger's KOH theorem and the strict unimodality of q-binomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4480","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":"sha256:99510f3c91870e4c923e19e51d67418b974d9e3d2f3a0eddc3bf0cb92180b68e","target":"record","created_at":"2026-05-18T02:18:31Z","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":"c09c119bf5c2ee5f99495b0c6edb2ca3a71d2d59c25c1bb9c392b4e16828c571","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-18T18:26:38Z","title_canon_sha256":"8b6aad0c48dd8c70beb148c809f44fb421acaec5cb41dfab962b8c3017214916"},"schema_version":"1.0","source":{"id":"1311.4480","kind":"arxiv","version":2}},"canonical_sha256":"d1c147d4a26f8c9ce61265a682aa66c93ae08d1cedf9415b3b516e6b6798fcb2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1c147d4a26f8c9ce61265a682aa66c93ae08d1cedf9415b3b516e6b6798fcb2","first_computed_at":"2026-05-18T02:18:31.731493Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:31.731493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MgVUKJ2KBX4/3+6/QG7l8G4daUxxOhVJbZz/CZKCiUTxdtbGDsrwcm1KzT1zc80IBg/x8MyY1aR1waUvZ3ARBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:31.731983Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.4480","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99510f3c91870e4c923e19e51d67418b974d9e3d2f3a0eddc3bf0cb92180b68e","sha256:65d466abd84207c605300661b29b55b73cab79bf76525632d46dfa213a09026a"],"state_sha256":"418a5cd21f2eb83562963b5e76143dec5eca21e070189d3442815c99984e4485"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k77/3phNbua+qx+7oujHGRTdhLDeOJU+WKnDK631I0TL4cAu6u/bReObCSFb2TCrYWheqhwIC5bvOmPb/vDrAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T03:48:31.880163Z","bundle_sha256":"136271dde56a707c67d5c8bc64ba1a428bf14f5cca42e2e9aaee456ddc472464"}}