{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:EFX4F3TV7EWIQUDUF3XNOFOXOB","short_pith_number":"pith:EFX4F3TV","canonical_record":{"source":{"id":"1604.08837","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-04-29T14:03:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"28bd511d4e29dbb3e2b748035600ce054a5de4a166db5eef711023332dca9e3d","abstract_canon_sha256":"b34d151648d3e8bfa0cbda8ac936b23cb14254de28d4406aeb87d780132db08c"},"schema_version":"1.0"},"canonical_sha256":"216fc2ee75f92c8850742eeed715d770758bc1c38bd813b86de8718a0d216e8c","source":{"kind":"arxiv","id":"1604.08837","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08837","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08837v1","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08837","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"EFX4F3TV7EWI","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"EFX4F3TV7EWIQUDU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"EFX4F3TV","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:EFX4F3TV7EWIQUDUF3XNOFOXOB","target":"record","payload":{"canonical_record":{"source":{"id":"1604.08837","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-04-29T14:03:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"28bd511d4e29dbb3e2b748035600ce054a5de4a166db5eef711023332dca9e3d","abstract_canon_sha256":"b34d151648d3e8bfa0cbda8ac936b23cb14254de28d4406aeb87d780132db08c"},"schema_version":"1.0"},"canonical_sha256":"216fc2ee75f92c8850742eeed715d770758bc1c38bd813b86de8718a0d216e8c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:17.700307Z","signature_b64":"Nu+Z8qUxmXVb4E5phxI/KahXiGGmOloWgGGzh02b8LHvP+0fb4VRVb96FKp64zTr9mMlx7/WYHs1Jxt48z+3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"216fc2ee75f92c8850742eeed715d770758bc1c38bd813b86de8718a0d216e8c","last_reissued_at":"2026-05-18T00:48:17.699630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:17.699630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.08837","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":"2026-05-18T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6L2OMPXrnPjesHa/dnUMsNtx7z07tYRiqC64SzbGKg1vo5/EdUOl+UcJodzsngDrUHXOrwgUhCmxqmF1jpRHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:55:43.350267Z"},"content_sha256":"06c7577ca8f15fad84ceb1118bc19548f49ecafede5a5219eae90e22a17261a5","schema_version":"1.0","event_id":"sha256:06c7577ca8f15fad84ceb1118bc19548f49ecafede5a5219eae90e22a17261a5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:EFX4F3TV7EWIQUDUF3XNOFOXOB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representations of symmetric groups with non-trivial determinant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Amritanshu Prasad, Arvind Ayyer, Steven Spallone","submitted_at":"2016-04-29T14:03:35Z","abstract_excerpt":"We give a closed formula for the number of partitions $\\lambda$ of $n$ such that the corresponding irreducible representation $V_\\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08837","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":"2026-05-18T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PRBJ5+OQWVMIqT9YezC+/c6I/D8lMkaR7oFUvJu7RbylI84a3yNwVumiFeGbaZqH2kFxxVmC1Kbh2KtLA01BBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:55:43.350691Z"},"content_sha256":"cc54cba3e357867258eacd3dd336256e60d4c49d651eb897f3622fd2280eec1e","schema_version":"1.0","event_id":"sha256:cc54cba3e357867258eacd3dd336256e60d4c49d651eb897f3622fd2280eec1e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/bundle.json","state_url":"https://pith.science/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/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":"2026-05-28T16:55:43Z","links":{"resolver":"https://pith.science/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB","bundle":"https://pith.science/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/bundle.json","state":"https://pith.science/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EFX4F3TV7EWIQUDUF3XNOFOXOB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:EFX4F3TV7EWIQUDUF3XNOFOXOB","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":"b34d151648d3e8bfa0cbda8ac936b23cb14254de28d4406aeb87d780132db08c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-04-29T14:03:35Z","title_canon_sha256":"28bd511d4e29dbb3e2b748035600ce054a5de4a166db5eef711023332dca9e3d"},"schema_version":"1.0","source":{"id":"1604.08837","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08837","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08837v1","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08837","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"EFX4F3TV7EWI","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"EFX4F3TV7EWIQUDU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"EFX4F3TV","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:cc54cba3e357867258eacd3dd336256e60d4c49d651eb897f3622fd2280eec1e","target":"graph","created_at":"2026-05-18T00:48:17Z","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 give a closed formula for the number of partitions $\\lambda$ of $n$ such that the corresponding irreducible representation $V_\\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.","authors_text":"Amritanshu Prasad, Arvind Ayyer, Steven Spallone","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-04-29T14:03:35Z","title":"Representations of symmetric groups with non-trivial determinant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08837","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":"sha256:06c7577ca8f15fad84ceb1118bc19548f49ecafede5a5219eae90e22a17261a5","target":"record","created_at":"2026-05-18T00:48:17Z","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":"b34d151648d3e8bfa0cbda8ac936b23cb14254de28d4406aeb87d780132db08c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-04-29T14:03:35Z","title_canon_sha256":"28bd511d4e29dbb3e2b748035600ce054a5de4a166db5eef711023332dca9e3d"},"schema_version":"1.0","source":{"id":"1604.08837","kind":"arxiv","version":1}},"canonical_sha256":"216fc2ee75f92c8850742eeed715d770758bc1c38bd813b86de8718a0d216e8c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"216fc2ee75f92c8850742eeed715d770758bc1c38bd813b86de8718a0d216e8c","first_computed_at":"2026-05-18T00:48:17.699630Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:17.699630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nu+Z8qUxmXVb4E5phxI/KahXiGGmOloWgGGzh02b8LHvP+0fb4VRVb96FKp64zTr9mMlx7/WYHs1Jxt48z+3BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:17.700307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.08837","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06c7577ca8f15fad84ceb1118bc19548f49ecafede5a5219eae90e22a17261a5","sha256:cc54cba3e357867258eacd3dd336256e60d4c49d651eb897f3622fd2280eec1e"],"state_sha256":"d1642db70dc6484dd69711a72ce4aa2bec00b511a06eb3af1f70faa4b944e787"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V3I9zwy4LJZgyHPwAtpZxFuLafaCX0HEHMuZYlhio77H33NlC+wFUg2b7PhBJJuWqkE48Jo+E7ltB4Phrp02Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:55:43.354433Z","bundle_sha256":"145e3ba412ea5fa5b5a709d938cd1578d1756589c8d694b69ae2a94a28534050"}}