{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OSU4XQVL3CIYKYFOQ75MXZY3PG","short_pith_number":"pith:OSU4XQVL","canonical_record":{"source":{"id":"1609.04208","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-14T10:32:53Z","cross_cats_sorted":[],"title_canon_sha256":"c51862e0fcf65866dd49f799599deab514da341bc2af2b14041779cbd4bd7f25","abstract_canon_sha256":"f3dbc26c245e6856223a4db5782583589e86710cc3ac3a5509bb1e0c257c96fc"},"schema_version":"1.0"},"canonical_sha256":"74a9cbc2abd8918560ae87facbe71b798171bb5b38f7119adc8ccbf84b94a9b0","source":{"kind":"arxiv","id":"1609.04208","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.04208","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1609.04208v1","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04208","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"OSU4XQVL3CIY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OSU4XQVL3CIYKYFO","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OSU4XQVL","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OSU4XQVL3CIYKYFOQ75MXZY3PG","target":"record","payload":{"canonical_record":{"source":{"id":"1609.04208","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-14T10:32:53Z","cross_cats_sorted":[],"title_canon_sha256":"c51862e0fcf65866dd49f799599deab514da341bc2af2b14041779cbd4bd7f25","abstract_canon_sha256":"f3dbc26c245e6856223a4db5782583589e86710cc3ac3a5509bb1e0c257c96fc"},"schema_version":"1.0"},"canonical_sha256":"74a9cbc2abd8918560ae87facbe71b798171bb5b38f7119adc8ccbf84b94a9b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:38.607358Z","signature_b64":"t0JSgpYJaYgnAkpjkDytZpoLNJkw8mJw4I73vc4yER3NI67szGg7kX2EmqKeg4NFmGHR6gOZEKMWQnjC0eFjDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74a9cbc2abd8918560ae87facbe71b798171bb5b38f7119adc8ccbf84b94a9b0","last_reissued_at":"2026-05-18T01:04:38.606699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:38.606699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.04208","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":"2026-05-18T01:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BgClD11DZCerT+Zs3xjs6nga8V0EjimVvM92GgTWKAyyfMSAF8Y+7osSp3yCvWxRuFiQPB8jF37HUv/BgAuGBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T14:48:55.870055Z"},"content_sha256":"c6af9c1b7f3e1fd18443dde167e056edc35c5a9262f5c4fb9ec02d19f1f12fd4","schema_version":"1.0","event_id":"sha256:c6af9c1b7f3e1fd18443dde167e056edc35c5a9262f5c4fb9ec02d19f1f12fd4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OSU4XQVL3CIYKYFOQ75MXZY3PG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $\\mu$-permanent, a new graph labeling, and a known integer sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Pereira, Carlos M. da Fonseca, Milica An{\\dj}eli\\'c","submitted_at":"2016-09-14T10:32:53Z","abstract_excerpt":"Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\\mu$, we define the polynomial $$P_\\mu(A)=\\sum_{\\sigma\\in S_n} a_{1\\sigma(1)}\\cdots a_{n\\sigma(n)}\\,\\mu^{\\ell(\\sigma)}\\; ,$$ as the $\\mu$-permanent of $A$, where $\\ell(\\sigma)$ is the number of inversions of the permutation $\\sigma$ in the symmetric group $S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $\\mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04208","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":"2026-05-18T01:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bqEVY9GsKKW7cxJcI2b4DbP77i2Ovydaf/+alBaq0g/XuG7adEpCNKRQ8y05ztymGjyReljAcfvHMSCvv2NLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T14:48:55.870420Z"},"content_sha256":"930441470e79fcd9dba04ce821ee89e2f61d2b53249ea645f1ca7217f3e295dc","schema_version":"1.0","event_id":"sha256:930441470e79fcd9dba04ce821ee89e2f61d2b53249ea645f1ca7217f3e295dc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/bundle.json","state_url":"https://pith.science/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/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":"2026-07-01T14:48:55Z","links":{"resolver":"https://pith.science/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG","bundle":"https://pith.science/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/bundle.json","state":"https://pith.science/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OSU4XQVL3CIYKYFOQ75MXZY3PG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OSU4XQVL3CIYKYFOQ75MXZY3PG","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":"f3dbc26c245e6856223a4db5782583589e86710cc3ac3a5509bb1e0c257c96fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-14T10:32:53Z","title_canon_sha256":"c51862e0fcf65866dd49f799599deab514da341bc2af2b14041779cbd4bd7f25"},"schema_version":"1.0","source":{"id":"1609.04208","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.04208","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1609.04208v1","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04208","created_at":"2026-05-18T01:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"OSU4XQVL3CIY","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OSU4XQVL3CIYKYFO","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OSU4XQVL","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:930441470e79fcd9dba04ce821ee89e2f61d2b53249ea645f1ca7217f3e295dc","target":"graph","created_at":"2026-05-18T01:04:38Z","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":"Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\\mu$, we define the polynomial $$P_\\mu(A)=\\sum_{\\sigma\\in S_n} a_{1\\sigma(1)}\\cdots a_{n\\sigma(n)}\\,\\mu^{\\ell(\\sigma)}\\; ,$$ as the $\\mu$-permanent of $A$, where $\\ell(\\sigma)$ is the number of inversions of the permutation $\\sigma$ in the symmetric group $S_n$. In this note, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $\\mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.","authors_text":"Ant\\'onio Pereira, Carlos M. da Fonseca, Milica An{\\dj}eli\\'c","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-14T10:32:53Z","title":"The $\\mu$-permanent, a new graph labeling, and a known integer sequence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04208","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:c6af9c1b7f3e1fd18443dde167e056edc35c5a9262f5c4fb9ec02d19f1f12fd4","target":"record","created_at":"2026-05-18T01:04:38Z","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":"f3dbc26c245e6856223a4db5782583589e86710cc3ac3a5509bb1e0c257c96fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-14T10:32:53Z","title_canon_sha256":"c51862e0fcf65866dd49f799599deab514da341bc2af2b14041779cbd4bd7f25"},"schema_version":"1.0","source":{"id":"1609.04208","kind":"arxiv","version":1}},"canonical_sha256":"74a9cbc2abd8918560ae87facbe71b798171bb5b38f7119adc8ccbf84b94a9b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74a9cbc2abd8918560ae87facbe71b798171bb5b38f7119adc8ccbf84b94a9b0","first_computed_at":"2026-05-18T01:04:38.606699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:38.606699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t0JSgpYJaYgnAkpjkDytZpoLNJkw8mJw4I73vc4yER3NI67szGg7kX2EmqKeg4NFmGHR6gOZEKMWQnjC0eFjDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:38.607358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.04208","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6af9c1b7f3e1fd18443dde167e056edc35c5a9262f5c4fb9ec02d19f1f12fd4","sha256:930441470e79fcd9dba04ce821ee89e2f61d2b53249ea645f1ca7217f3e295dc"],"state_sha256":"a1725401f399b8ee121a913894f2aa2c089ae1a73ab56847a7b0acf74933f05c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TDzEgrVDEGr7gzAaJiDBwUOzejJTXApQ+WmX8VW1gYgeZPcaJGoXYGV5q9GtvTo3TLKW1crW1/zYVXnvPNA8Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T14:48:55.872325Z","bundle_sha256":"90f78166f365549fff0f8126e688472270199769f436373c50da98b01e0b072b"}}