{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:BF5Z4C457QKJNYL46XBKTVNCPD","short_pith_number":"pith:BF5Z4C45","canonical_record":{"source":{"id":"1804.05933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-16T20:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"33d9ddbde083f52218962f2b52f289523858c4324534f6233c6dc7dd46e64a61","abstract_canon_sha256":"094b29966a31afd17eb10c69ddedf16a51e1e112fbe121809afb58fb6f2b33bd"},"schema_version":"1.0"},"canonical_sha256":"097b9e0b9dfc1496e17cf5c2a9d5a278db5e0912d731c1b16c43fffab3815582","source":{"kind":"arxiv","id":"1804.05933","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.05933","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"arxiv_version","alias_value":"1804.05933v1","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.05933","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"pith_short_12","alias_value":"BF5Z4C457QKJ","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BF5Z4C457QKJNYL4","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BF5Z4C45","created_at":"2026-05-18T12:32:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:BF5Z4C457QKJNYL46XBKTVNCPD","target":"record","payload":{"canonical_record":{"source":{"id":"1804.05933","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-16T20:46:38Z","cross_cats_sorted":[],"title_canon_sha256":"33d9ddbde083f52218962f2b52f289523858c4324534f6233c6dc7dd46e64a61","abstract_canon_sha256":"094b29966a31afd17eb10c69ddedf16a51e1e112fbe121809afb58fb6f2b33bd"},"schema_version":"1.0"},"canonical_sha256":"097b9e0b9dfc1496e17cf5c2a9d5a278db5e0912d731c1b16c43fffab3815582","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:22.302990Z","signature_b64":"2ev5/ZNbHugn05fucxEbg0pXa7iTuajwmq5Dvs3N3bvl6vQUbx0Qsdpu6icujbINYPobJzmQithermCcfvUBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"097b9e0b9dfc1496e17cf5c2a9d5a278db5e0912d731c1b16c43fffab3815582","last_reissued_at":"2026-05-18T00:18:22.302199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:22.302199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.05933","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":"2026-05-18T00:18:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hFwcQ0OUAn48QpRve5phlirzwH+LU8CUxYm1WbNShkebSNM/oh8uSG9xcTajDNLIpAvA1GQM0SLRtD/o2zRJAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T20:44:31.921225Z"},"content_sha256":"6b6b09606d21f433a7de93a377e9600f3c83190c092646c7fdfeb6d89604523a","schema_version":"1.0","event_id":"sha256:6b6b09606d21f433a7de93a377e9600f3c83190c092646c7fdfeb6d89604523a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:BF5Z4C457QKJNYL46XBKTVNCPD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bryan Ek","submitted_at":"2018-04-16T20:46:38Z","abstract_excerpt":"The main theme of this dissertation is retooling methods to work for different situations.\n  I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.05933","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":"2026-05-18T00:18:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IJcyV8Y5xvTaB9v5+PLWdTQE0+s/Gu6ujhy1aOIn9FYj9phFf+a7rTJNrWBMYy0jbumJqmumEDubbkYJ6Sn/AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T20:44:31.921586Z"},"content_sha256":"16999fcaeb16de742efe0f277af2a2c78d731a89042ce7718b238bbf987f0ea2","schema_version":"1.0","event_id":"sha256:16999fcaeb16de742efe0f277af2a2c78d731a89042ce7718b238bbf987f0ea2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BF5Z4C457QKJNYL46XBKTVNCPD/bundle.json","state_url":"https://pith.science/pith/BF5Z4C457QKJNYL46XBKTVNCPD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BF5Z4C457QKJNYL46XBKTVNCPD/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":"2026-06-03T20:44:31Z","links":{"resolver":"https://pith.science/pith/BF5Z4C457QKJNYL46XBKTVNCPD","bundle":"https://pith.science/pith/BF5Z4C457QKJNYL46XBKTVNCPD/bundle.json","state":"https://pith.science/pith/BF5Z4C457QKJNYL46XBKTVNCPD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BF5Z4C457QKJNYL46XBKTVNCPD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BF5Z4C457QKJNYL46XBKTVNCPD","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":"094b29966a31afd17eb10c69ddedf16a51e1e112fbe121809afb58fb6f2b33bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-16T20:46:38Z","title_canon_sha256":"33d9ddbde083f52218962f2b52f289523858c4324534f6233c6dc7dd46e64a61"},"schema_version":"1.0","source":{"id":"1804.05933","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.05933","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"arxiv_version","alias_value":"1804.05933v1","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.05933","created_at":"2026-05-18T00:18:22Z"},{"alias_kind":"pith_short_12","alias_value":"BF5Z4C457QKJ","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BF5Z4C457QKJNYL4","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BF5Z4C45","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:16999fcaeb16de742efe0f277af2a2c78d731a89042ce7718b238bbf987f0ea2","target":"graph","created_at":"2026-05-18T00:18:22Z","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":"The main theme of this dissertation is retooling methods to work for different situations.\n  I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded p","authors_text":"Bryan Ek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-16T20:46:38Z","title":"Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.05933","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":"sha256:6b6b09606d21f433a7de93a377e9600f3c83190c092646c7fdfeb6d89604523a","target":"record","created_at":"2026-05-18T00:18:22Z","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":"094b29966a31afd17eb10c69ddedf16a51e1e112fbe121809afb58fb6f2b33bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-16T20:46:38Z","title_canon_sha256":"33d9ddbde083f52218962f2b52f289523858c4324534f6233c6dc7dd46e64a61"},"schema_version":"1.0","source":{"id":"1804.05933","kind":"arxiv","version":1}},"canonical_sha256":"097b9e0b9dfc1496e17cf5c2a9d5a278db5e0912d731c1b16c43fffab3815582","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"097b9e0b9dfc1496e17cf5c2a9d5a278db5e0912d731c1b16c43fffab3815582","first_computed_at":"2026-05-18T00:18:22.302199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:22.302199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2ev5/ZNbHugn05fucxEbg0pXa7iTuajwmq5Dvs3N3bvl6vQUbx0Qsdpu6icujbINYPobJzmQithermCcfvUBCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:22.302990Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.05933","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6b6b09606d21f433a7de93a377e9600f3c83190c092646c7fdfeb6d89604523a","sha256:16999fcaeb16de742efe0f277af2a2c78d731a89042ce7718b238bbf987f0ea2"],"state_sha256":"b0b295aa2df1a2ffa39a1eb21e4071e68c251ce832d671912d3d0fb99133d566"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"24+7GfmAT4ibojdF+JBAlUo84rTM5K/km/krfdKrT20czIrxunyUJmWOpwj03JxPDXjisxGXnGH7lTzaHt80Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T20:44:31.923591Z","bundle_sha256":"980f158c12ff46710651a128013ee3c2cc1f20b5a70c752d1ac44ea9c7dde1b1"}}