{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:MFG725AEALUGPWL63E5MJKH6FX","short_pith_number":"pith:MFG725AE","canonical_record":{"source":{"id":"0906.5573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-30T16:14:29Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a21d2bc2bb905d201e7ec623b4c2386f9c432159a0deefb664ed53f54ff27663","abstract_canon_sha256":"b5da6260394ffe8c503b2ab464ff3d8f6aa76b9fd7a75f74e55a091fb44607bc"},"schema_version":"1.0"},"canonical_sha256":"614dfd740402e867d97ed93ac4a8fe2dd6b5659ff4dd5458d0421240e0600f6a","source":{"kind":"arxiv","id":"0906.5573","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.5573","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"arxiv_version","alias_value":"0906.5573v2","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.5573","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"pith_short_12","alias_value":"MFG725AEALUG","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"MFG725AEALUGPWL6","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"MFG725AE","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:MFG725AEALUGPWL63E5MJKH6FX","target":"record","payload":{"canonical_record":{"source":{"id":"0906.5573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-30T16:14:29Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"a21d2bc2bb905d201e7ec623b4c2386f9c432159a0deefb664ed53f54ff27663","abstract_canon_sha256":"b5da6260394ffe8c503b2ab464ff3d8f6aa76b9fd7a75f74e55a091fb44607bc"},"schema_version":"1.0"},"canonical_sha256":"614dfd740402e867d97ed93ac4a8fe2dd6b5659ff4dd5458d0421240e0600f6a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:53.050116Z","signature_b64":"yV7PPuzygXNgi+751NoTpftyA9W7UDA4SC+FmdZgKuA/N35du8874fENWZGmdqQSqn3oFrxGJhPgev6+XkisAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"614dfd740402e867d97ed93ac4a8fe2dd6b5659ff4dd5458d0421240e0600f6a","last_reissued_at":"2026-05-18T03:07:53.049573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:53.049573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.5573","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":"2026-05-18T03:07:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mwhrLtxRtW8isY7udGSVLWPaKaVmn5Xu5fODQTmTaW9uA3/gIiBNEoXXzleJWAIuM8jOKe7bLEUaaNQEFjvvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:39:58.028917Z"},"content_sha256":"29ef52c529a2aebc6e9d519d37a1d77757ce2c0bb804004d1381cc84d7853a16","schema_version":"1.0","event_id":"sha256:29ef52c529a2aebc6e9d519d37a1d77757ce2c0bb804004d1381cc84d7853a16"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:MFG725AEALUGPWL63E5MJKH6FX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetrically Constrained Compositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Carla D. Savage, Ira M. Gessel, Matthias Beck, Sunyoung Lee","submitted_at":"2009-06-30T16:14:29Z","abstract_excerpt":"Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \\geq 1$, a symmetrically constrained composition $\\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints\n  ${\\sum_{i=1}^n a_i \\lambda_{\\pi(i)} \\geq 0 : \\pi \\in S_n}$. We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point enumeration."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.5573","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":"2026-05-18T03:07:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"quaE5p2mJ0VS/mTvJhgkc4gzjLq6ws9c9o5AiZeRUlPZSe2B7zxd8PcV4se2F61zUAzjg5zSvexiB0vcRQMTDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:39:58.029252Z"},"content_sha256":"9d36987a3ac61b81ba788657d5d4049d08bbfeb94423a011ff174753ea1833b3","schema_version":"1.0","event_id":"sha256:9d36987a3ac61b81ba788657d5d4049d08bbfeb94423a011ff174753ea1833b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MFG725AEALUGPWL63E5MJKH6FX/bundle.json","state_url":"https://pith.science/pith/MFG725AEALUGPWL63E5MJKH6FX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MFG725AEALUGPWL63E5MJKH6FX/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":"2026-06-03T04:39:58Z","links":{"resolver":"https://pith.science/pith/MFG725AEALUGPWL63E5MJKH6FX","bundle":"https://pith.science/pith/MFG725AEALUGPWL63E5MJKH6FX/bundle.json","state":"https://pith.science/pith/MFG725AEALUGPWL63E5MJKH6FX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MFG725AEALUGPWL63E5MJKH6FX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:MFG725AEALUGPWL63E5MJKH6FX","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":"b5da6260394ffe8c503b2ab464ff3d8f6aa76b9fd7a75f74e55a091fb44607bc","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-30T16:14:29Z","title_canon_sha256":"a21d2bc2bb905d201e7ec623b4c2386f9c432159a0deefb664ed53f54ff27663"},"schema_version":"1.0","source":{"id":"0906.5573","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.5573","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"arxiv_version","alias_value":"0906.5573v2","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.5573","created_at":"2026-05-18T03:07:53Z"},{"alias_kind":"pith_short_12","alias_value":"MFG725AEALUG","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"MFG725AEALUGPWL6","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"MFG725AE","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:9d36987a3ac61b81ba788657d5d4049d08bbfeb94423a011ff174753ea1833b3","target":"graph","created_at":"2026-05-18T03:07:53Z","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":"Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \\geq 1$, a symmetrically constrained composition $\\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints\n  ${\\sum_{i=1}^n a_i \\lambda_{\\pi(i)} \\geq 0 : \\pi \\in S_n}$. We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point enumeration.","authors_text":"Carla D. Savage, Ira M. Gessel, Matthias Beck, Sunyoung Lee","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-30T16:14:29Z","title":"Symmetrically Constrained Compositions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.5573","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":"sha256:29ef52c529a2aebc6e9d519d37a1d77757ce2c0bb804004d1381cc84d7853a16","target":"record","created_at":"2026-05-18T03:07:53Z","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":"b5da6260394ffe8c503b2ab464ff3d8f6aa76b9fd7a75f74e55a091fb44607bc","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-06-30T16:14:29Z","title_canon_sha256":"a21d2bc2bb905d201e7ec623b4c2386f9c432159a0deefb664ed53f54ff27663"},"schema_version":"1.0","source":{"id":"0906.5573","kind":"arxiv","version":2}},"canonical_sha256":"614dfd740402e867d97ed93ac4a8fe2dd6b5659ff4dd5458d0421240e0600f6a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"614dfd740402e867d97ed93ac4a8fe2dd6b5659ff4dd5458d0421240e0600f6a","first_computed_at":"2026-05-18T03:07:53.049573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:53.049573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yV7PPuzygXNgi+751NoTpftyA9W7UDA4SC+FmdZgKuA/N35du8874fENWZGmdqQSqn3oFrxGJhPgev6+XkisAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:53.050116Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.5573","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29ef52c529a2aebc6e9d519d37a1d77757ce2c0bb804004d1381cc84d7853a16","sha256:9d36987a3ac61b81ba788657d5d4049d08bbfeb94423a011ff174753ea1833b3"],"state_sha256":"3e912406c9e64664c0927528b749e49ac9bb16a249f1339f03a37a2ce317f6b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nPRZ985Y1+R6FFx46atuUsvD5EW+1uNh30Z5Ecl5ebb3hZCVz7C3uelYFY5tTyWC9AzlcvsrvJ+ZZHVMDzs+Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:39:58.031154Z","bundle_sha256":"f89ba02adfe1e1b1f70ff366ff1f6f1ed48f39920f5ca7b5c5fdce5b6cf2a8bb"}}