{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:2XPCBIYVPQ6OO7REYV4NAUFIKO","short_pith_number":"pith:2XPCBIYV","canonical_record":{"source":{"id":"1109.3353","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-15T14:06:17Z","cross_cats_sorted":[],"title_canon_sha256":"57841588b901c4e5d174234e4155fab2b4befd0e3f02bc8b6b0d96c293abeb52","abstract_canon_sha256":"228d0f939546110e5d953ca8f3aa737e3083093b945cbb8fcea0f650483419d9"},"schema_version":"1.0"},"canonical_sha256":"d5de20a3157c3ce77e24c578d050a853a343f1dae5d121bef1ef9e1c9a303d59","source":{"kind":"arxiv","id":"1109.3353","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3353","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3353v3","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3353","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"pith_short_12","alias_value":"2XPCBIYVPQ6O","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2XPCBIYVPQ6OO7RE","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2XPCBIYV","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:2XPCBIYVPQ6OO7REYV4NAUFIKO","target":"record","payload":{"canonical_record":{"source":{"id":"1109.3353","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-15T14:06:17Z","cross_cats_sorted":[],"title_canon_sha256":"57841588b901c4e5d174234e4155fab2b4befd0e3f02bc8b6b0d96c293abeb52","abstract_canon_sha256":"228d0f939546110e5d953ca8f3aa737e3083093b945cbb8fcea0f650483419d9"},"schema_version":"1.0"},"canonical_sha256":"d5de20a3157c3ce77e24c578d050a853a343f1dae5d121bef1ef9e1c9a303d59","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:24.274590Z","signature_b64":"l9RdG2Vz/e5kc3unFZ6oBq4Au9yIVuH1W7fSP3SzGKntu22o/OHt8gAnP//3DNXfE1OUotco7PbfLOrEzmt3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5de20a3157c3ce77e24c578d050a853a343f1dae5d121bef1ef9e1c9a303d59","last_reissued_at":"2026-05-18T03:11:24.273863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:24.273863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.3353","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:11:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xga82/upuZ2qiKT2ogRnri/IUyhGV1nAJkuSIy/b63gHVz9Jb+mBhYVkuJBQlAco5ujvxmthCXykCG2eV4VMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:52:22.530856Z"},"content_sha256":"d499af33ac20938c155acc31c5a37a2261021ab1026b89de5372d3f6e64685f4","schema_version":"1.0","event_id":"sha256:d499af33ac20938c155acc31c5a37a2261021ab1026b89de5372d3f6e64685f4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:2XPCBIYVPQ6OO7REYV4NAUFIKO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Euler-Mahonian Statistics via Polyhedral Geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin Braun, Matthias Beck","submitted_at":"2011-09-15T14:06:17Z","abstract_excerpt":"A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate generating function identity encoding these statistics. We use techniques from polyhedral geometry to establish new multivariate generalizations for many of the known Euler--Mahonian distributions. The original bivariate distributions are then straightforward specializations of these multivariate identities. A consequence of these new techniques are bijective pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3353","kind":"arxiv","version":3},"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":"2026-05-18T03:11:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Md4xLtVSgA9JwBocaUmRXdTmk8xwcTLh17Q8Ma8w45XCwEKbPPgcD+ZoFcykEDHFi2ZTcS8EHhNogEC/GJQOAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:52:22.531204Z"},"content_sha256":"097151e175d4715d0eab9d0e3b28ffb9dc971d2358d7ace8d7e6eee88aac86f6","schema_version":"1.0","event_id":"sha256:097151e175d4715d0eab9d0e3b28ffb9dc971d2358d7ace8d7e6eee88aac86f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/bundle.json","state_url":"https://pith.science/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/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":"2026-06-03T06:52:22Z","links":{"resolver":"https://pith.science/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO","bundle":"https://pith.science/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/bundle.json","state":"https://pith.science/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2XPCBIYVPQ6OO7REYV4NAUFIKO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2XPCBIYVPQ6OO7REYV4NAUFIKO","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":"228d0f939546110e5d953ca8f3aa737e3083093b945cbb8fcea0f650483419d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-15T14:06:17Z","title_canon_sha256":"57841588b901c4e5d174234e4155fab2b4befd0e3f02bc8b6b0d96c293abeb52"},"schema_version":"1.0","source":{"id":"1109.3353","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3353","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3353v3","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3353","created_at":"2026-05-18T03:11:24Z"},{"alias_kind":"pith_short_12","alias_value":"2XPCBIYVPQ6O","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2XPCBIYVPQ6OO7RE","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2XPCBIYV","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:097151e175d4715d0eab9d0e3b28ffb9dc971d2358d7ace8d7e6eee88aac86f6","target":"graph","created_at":"2026-05-18T03:11:24Z","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 variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate generating function identity encoding these statistics. We use techniques from polyhedral geometry to establish new multivariate generalizations for many of the known Euler--Mahonian distributions. The original bivariate distributions are then straightforward specializations of these multivariate identities. A consequence of these new techniques are bijective pr","authors_text":"Benjamin Braun, Matthias Beck","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-15T14:06:17Z","title":"Euler-Mahonian Statistics via Polyhedral Geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3353","kind":"arxiv","version":3},"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:d499af33ac20938c155acc31c5a37a2261021ab1026b89de5372d3f6e64685f4","target":"record","created_at":"2026-05-18T03:11:24Z","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":"228d0f939546110e5d953ca8f3aa737e3083093b945cbb8fcea0f650483419d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-15T14:06:17Z","title_canon_sha256":"57841588b901c4e5d174234e4155fab2b4befd0e3f02bc8b6b0d96c293abeb52"},"schema_version":"1.0","source":{"id":"1109.3353","kind":"arxiv","version":3}},"canonical_sha256":"d5de20a3157c3ce77e24c578d050a853a343f1dae5d121bef1ef9e1c9a303d59","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d5de20a3157c3ce77e24c578d050a853a343f1dae5d121bef1ef9e1c9a303d59","first_computed_at":"2026-05-18T03:11:24.273863Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:24.273863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l9RdG2Vz/e5kc3unFZ6oBq4Au9yIVuH1W7fSP3SzGKntu22o/OHt8gAnP//3DNXfE1OUotco7PbfLOrEzmt3BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:24.274590Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.3353","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d499af33ac20938c155acc31c5a37a2261021ab1026b89de5372d3f6e64685f4","sha256:097151e175d4715d0eab9d0e3b28ffb9dc971d2358d7ace8d7e6eee88aac86f6"],"state_sha256":"0f477e8595dd30a75b0b22f7082d8d9a0298b0f2714ff59e4494a8b27fa47b6a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DIUAjjBcowYgHtGlnnqTBjdqnIQ6LX/0SEwjW8iJQaj8m/R7H0HVmna65QIu6hvGSVBrHTLDHfxhvo+Zg3kLBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T06:52:22.533138Z","bundle_sha256":"b4fe06f41af6d073e92a13dd05ac9839da037c10991dd64c05a975cd34232bec"}}